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Theorem naddcllem 8675
Description: Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
Assertion
Ref Expression
naddcllem ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem naddcllem
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑓 𝑡 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7416 . . . 4 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
21eleq1d 2819 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) ∈ On ↔ (𝑐 +no 𝑏) ∈ On))
3 sneq 4639 . . . . . . . . . 10 (𝑎 = 𝑐 → {𝑎} = {𝑐})
43xpeq1d 5706 . . . . . . . . 9 (𝑎 = 𝑐 → ({𝑎} × 𝑏) = ({𝑐} × 𝑏))
54imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ ({𝑎} × 𝑏)) = ( +no “ ({𝑐} × 𝑏)))
65sseq1d 4014 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥))
7 xpeq1 5691 . . . . . . . . 9 (𝑎 = 𝑐 → (𝑎 × {𝑏}) = (𝑐 × {𝑏}))
87imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ (𝑎 × {𝑏})) = ( +no “ (𝑐 × {𝑏})))
98sseq1d 4014 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥))
106, 9anbi12d 632 . . . . . 6 (𝑎 = 𝑐 → ((( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)))
1110rabbidv 3441 . . . . 5 (𝑎 = 𝑐 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)})
1211inteqd 4956 . . . 4 (𝑎 = 𝑐 {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)})
131, 12eqeq12d 2749 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} ↔ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}))
142, 13anbi12d 632 . 2 (𝑎 = 𝑐 → (((𝑎 +no 𝑏) ∈ On ∧ (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)}) ↔ ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)})))
15 oveq2 7417 . . . 4 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
1615eleq1d 2819 . . 3 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) ∈ On ↔ (𝑐 +no 𝑑) ∈ On))
17 xpeq2 5698 . . . . . . . . 9 (𝑏 = 𝑑 → ({𝑐} × 𝑏) = ({𝑐} × 𝑑))
1817imaeq2d 6060 . . . . . . . 8 (𝑏 = 𝑑 → ( +no “ ({𝑐} × 𝑏)) = ( +no “ ({𝑐} × 𝑑)))
1918sseq1d 4014 . . . . . . 7 (𝑏 = 𝑑 → (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥))
20 sneq 4639 . . . . . . . . . 10 (𝑏 = 𝑑 → {𝑏} = {𝑑})
2120xpeq2d 5707 . . . . . . . . 9 (𝑏 = 𝑑 → (𝑐 × {𝑏}) = (𝑐 × {𝑑}))
2221imaeq2d 6060 . . . . . . . 8 (𝑏 = 𝑑 → ( +no “ (𝑐 × {𝑏})) = ( +no “ (𝑐 × {𝑑})))
2322sseq1d 4014 . . . . . . 7 (𝑏 = 𝑑 → (( +no “ (𝑐 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥))
2419, 23anbi12d 632 . . . . . 6 (𝑏 = 𝑑 → ((( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)))
2524rabbidv 3441 . . . . 5 (𝑏 = 𝑑 → {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
2625inteqd 4956 . . . 4 (𝑏 = 𝑑 {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
2715, 26eqeq12d 2749 . . 3 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)} ↔ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}))
2816, 27anbi12d 632 . 2 (𝑏 = 𝑑 → (((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) ↔ ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})))
29 oveq1 7416 . . . 4 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
3029eleq1d 2819 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) ∈ On ↔ (𝑐 +no 𝑑) ∈ On))
313xpeq1d 5706 . . . . . . . . 9 (𝑎 = 𝑐 → ({𝑎} × 𝑑) = ({𝑐} × 𝑑))
3231imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ ({𝑎} × 𝑑)) = ( +no “ ({𝑐} × 𝑑)))
3332sseq1d 4014 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥))
34 xpeq1 5691 . . . . . . . . 9 (𝑎 = 𝑐 → (𝑎 × {𝑑}) = (𝑐 × {𝑑}))
3534imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ (𝑎 × {𝑑})) = ( +no “ (𝑐 × {𝑑})))
3635sseq1d 4014 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ (𝑎 × {𝑑})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥))
3733, 36anbi12d 632 . . . . . 6 (𝑎 = 𝑐 → ((( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)))
3837rabbidv 3441 . . . . 5 (𝑎 = 𝑐 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
3938inteqd 4956 . . . 4 (𝑎 = 𝑐 {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
4029, 39eqeq12d 2749 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)} ↔ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}))
4130, 40anbi12d 632 . 2 (𝑎 = 𝑐 → (((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)}) ↔ ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})))
42 oveq1 7416 . . . 4 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
4342eleq1d 2819 . . 3 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) ∈ On ↔ (𝐴 +no 𝑏) ∈ On))
44 sneq 4639 . . . . . . . . . 10 (𝑎 = 𝐴 → {𝑎} = {𝐴})
4544xpeq1d 5706 . . . . . . . . 9 (𝑎 = 𝐴 → ({𝑎} × 𝑏) = ({𝐴} × 𝑏))
4645imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝐴 → ( +no “ ({𝑎} × 𝑏)) = ( +no “ ({𝐴} × 𝑏)))
4746sseq1d 4014 . . . . . . 7 (𝑎 = 𝐴 → (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥))
48 xpeq1 5691 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎 × {𝑏}) = (𝐴 × {𝑏}))
4948imaeq2d 6060 . . . . . . . 8 (𝑎 = 𝐴 → ( +no “ (𝑎 × {𝑏})) = ( +no “ (𝐴 × {𝑏})))
5049sseq1d 4014 . . . . . . 7 (𝑎 = 𝐴 → (( +no “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥))
5147, 50anbi12d 632 . . . . . 6 (𝑎 = 𝐴 → ((( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)))
5251rabbidv 3441 . . . . 5 (𝑎 = 𝐴 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)})
5352inteqd 4956 . . . 4 (𝑎 = 𝐴 {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)})
5442, 53eqeq12d 2749 . . 3 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} ↔ (𝐴 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)}))
5543, 54anbi12d 632 . 2 (𝑎 = 𝐴 → (((𝑎 +no 𝑏) ∈ On ∧ (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)}) ↔ ((𝐴 +no 𝑏) ∈ On ∧ (𝐴 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)})))
56 oveq2 7417 . . . 4 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
5756eleq1d 2819 . . 3 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) ∈ On ↔ (𝐴 +no 𝐵) ∈ On))
58 xpeq2 5698 . . . . . . . . 9 (𝑏 = 𝐵 → ({𝐴} × 𝑏) = ({𝐴} × 𝐵))
5958imaeq2d 6060 . . . . . . . 8 (𝑏 = 𝐵 → ( +no “ ({𝐴} × 𝑏)) = ( +no “ ({𝐴} × 𝐵)))
6059sseq1d 4014 . . . . . . 7 (𝑏 = 𝐵 → (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥))
61 sneq 4639 . . . . . . . . . 10 (𝑏 = 𝐵 → {𝑏} = {𝐵})
6261xpeq2d 5707 . . . . . . . . 9 (𝑏 = 𝐵 → (𝐴 × {𝑏}) = (𝐴 × {𝐵}))
6362imaeq2d 6060 . . . . . . . 8 (𝑏 = 𝐵 → ( +no “ (𝐴 × {𝑏})) = ( +no “ (𝐴 × {𝐵})))
6463sseq1d 4014 . . . . . . 7 (𝑏 = 𝐵 → (( +no “ (𝐴 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥))
6560, 64anbi12d 632 . . . . . 6 (𝑏 = 𝐵 → ((( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)))
6665rabbidv 3441 . . . . 5 (𝑏 = 𝐵 → {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
6766inteqd 4956 . . . 4 (𝑏 = 𝐵 {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
6856, 67eqeq12d 2749 . . 3 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)} ↔ (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
6957, 68anbi12d 632 . 2 (𝑏 = 𝐵 → (((𝐴 +no 𝑏) ∈ On ∧ (𝐴 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)}) ↔ ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})))
70 simpl 484 . . . . . 6 (((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) → (𝑐 +no 𝑏) ∈ On)
7170ralimi 3084 . . . . 5 (∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) → ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On)
72713ad2ant2 1135 . . . 4 ((∀𝑐𝑎𝑑𝑏 ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}) ∧ ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) ∧ ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)})) → ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On)
73 simpl 484 . . . . . 6 (((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)}) → (𝑎 +no 𝑑) ∈ On)
7473ralimi 3084 . . . . 5 (∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)}) → ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
75743ad2ant3 1136 . . . 4 ((∀𝑐𝑎𝑑𝑏 ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}) ∧ ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) ∧ ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)})) → ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
7672, 75jca 513 . . 3 ((∀𝑐𝑎𝑑𝑏 ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}) ∧ ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) ∧ ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)})) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On))
77 df-nadd 8665 . . . . . . . . 9 +no = frecs({⟨𝑝, 𝑞⟩ ∣ (𝑝 ∈ (On × On) ∧ 𝑞 ∈ (On × On) ∧ (((1st𝑝) E (1st𝑞) ∨ (1st𝑝) = (1st𝑞)) ∧ ((2nd𝑝) E (2nd𝑞) ∨ (2nd𝑝) = (2nd𝑞)) ∧ 𝑝𝑞))}, (On × On), (𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)}))
7877on2recsov 8667 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = (⟨𝑎, 𝑏⟩(𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)})( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))))
7978adantr 482 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 +no 𝑏) = (⟨𝑎, 𝑏⟩(𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)})( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))))
80 opex 5465 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
81 naddfn 8674 . . . . . . . . . 10 +no Fn (On × On)
82 fnfun 6650 . . . . . . . . . 10 ( +no Fn (On × On) → Fun +no )
8381, 82ax-mp 5 . . . . . . . . 9 Fun +no
84 vex 3479 . . . . . . . . . . . 12 𝑎 ∈ V
8584sucex 7794 . . . . . . . . . . 11 suc 𝑎 ∈ V
86 vex 3479 . . . . . . . . . . . 12 𝑏 ∈ V
8786sucex 7794 . . . . . . . . . . 11 suc 𝑏 ∈ V
8885, 87xpex 7740 . . . . . . . . . 10 (suc 𝑎 × suc 𝑏) ∈ V
8988difexi 5329 . . . . . . . . 9 ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ∈ V
90 resfunexg 7217 . . . . . . . . 9 ((Fun +no ∧ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ∈ V) → ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) ∈ V)
9183, 89, 90mp2an 691 . . . . . . . 8 ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) ∈ V
92 eloni 6375 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ On → Ord 𝑏)
9392ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → Ord 𝑏)
94 ordirr 6383 . . . . . . . . . . . . . . . . . 18 (Ord 𝑏 → ¬ 𝑏𝑏)
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ 𝑏𝑏)
9695olcd 873 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (¬ 𝑎 ∈ {𝑎} ∨ ¬ 𝑏𝑏))
97 ianor 981 . . . . . . . . . . . . . . . . 17 (¬ (𝑎 ∈ {𝑎} ∧ 𝑏𝑏) ↔ (¬ 𝑎 ∈ {𝑎} ∨ ¬ 𝑏𝑏))
98 opelxp 5713 . . . . . . . . . . . . . . . . 17 (⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏) ↔ (𝑎 ∈ {𝑎} ∧ 𝑏𝑏))
9997, 98xchnxbir 333 . . . . . . . . . . . . . . . 16 (¬ ⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏) ↔ (¬ 𝑎 ∈ {𝑎} ∨ ¬ 𝑏𝑏))
10096, 99sylibr 233 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ ⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏))
10184sucid 6447 . . . . . . . . . . . . . . . . . 18 𝑎 ∈ suc 𝑎
102 snssi 4812 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ suc 𝑎 → {𝑎} ⊆ suc 𝑎)
103101, 102ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑎} ⊆ suc 𝑎
104 sssucid 6445 . . . . . . . . . . . . . . . . 17 𝑏 ⊆ suc 𝑏
105 xpss12 5692 . . . . . . . . . . . . . . . . 17 (({𝑎} ⊆ suc 𝑎𝑏 ⊆ suc 𝑏) → ({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏))
106103, 104, 105mp2an 691 . . . . . . . . . . . . . . . 16 ({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏)
107 ssdifsn 4792 . . . . . . . . . . . . . . . 16 (({𝑎} × 𝑏) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ (({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏) ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏)))
108106, 107mpbiran 708 . . . . . . . . . . . . . . 15 (({𝑎} × 𝑏) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏))
109100, 108sylibr 233 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ({𝑎} × 𝑏) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))
110 resima2 6017 . . . . . . . . . . . . . 14 (({𝑎} × 𝑏) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) → (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) = ( +no “ ({𝑎} × 𝑏)))
111109, 110syl 17 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) = ( +no “ ({𝑎} × 𝑏)))
112111sseq1d 4014 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥))
113 eloni 6375 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ On → Ord 𝑎)
114113ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → Ord 𝑎)
115 ordirr 6383 . . . . . . . . . . . . . . . . . 18 (Ord 𝑎 → ¬ 𝑎𝑎)
116114, 115syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ 𝑎𝑎)
117116orcd 872 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ {𝑏}))
118 ianor 981 . . . . . . . . . . . . . . . . 17 (¬ (𝑎𝑎𝑏 ∈ {𝑏}) ↔ (¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ {𝑏}))
119 opelxp 5713 . . . . . . . . . . . . . . . . 17 (⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏}) ↔ (𝑎𝑎𝑏 ∈ {𝑏}))
120118, 119xchnxbir 333 . . . . . . . . . . . . . . . 16 (¬ ⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏}) ↔ (¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ {𝑏}))
121117, 120sylibr 233 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ ⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏}))
122 sssucid 6445 . . . . . . . . . . . . . . . . 17 𝑎 ⊆ suc 𝑎
12386sucid 6447 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ suc 𝑏
124 snssi 4812 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ suc 𝑏 → {𝑏} ⊆ suc 𝑏)
125123, 124ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑏} ⊆ suc 𝑏
126 xpss12 5692 . . . . . . . . . . . . . . . . 17 ((𝑎 ⊆ suc 𝑎 ∧ {𝑏} ⊆ suc 𝑏) → (𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏))
127122, 125, 126mp2an 691 . . . . . . . . . . . . . . . 16 (𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏)
128 ssdifsn 4792 . . . . . . . . . . . . . . . 16 ((𝑎 × {𝑏}) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ ((𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏) ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏})))
129127, 128mpbiran 708 . . . . . . . . . . . . . . 15 ((𝑎 × {𝑏}) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏}))
130121, 129sylibr 233 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 × {𝑏}) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))
131 resima2 6017 . . . . . . . . . . . . . 14 ((𝑎 × {𝑏}) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) → (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) = ( +no “ (𝑎 × {𝑏})))
132130, 131syl 17 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) = ( +no “ (𝑎 × {𝑏})))
133132sseq1d 4014 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥))
134112, 133anbi12d 632 . . . . . . . . . . 11 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)))
135134rabbidv 3441 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)})
136135inteqd 4956 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)})
137 simprr 772 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
138 oveq1 7416 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑎 → (𝑡 +no 𝑑) = (𝑎 +no 𝑑))
139138eleq1d 2819 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑎 → ((𝑡 +no 𝑑) ∈ On ↔ (𝑎 +no 𝑑) ∈ On))
140139ralbidv 3178 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑎 → (∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On ↔ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On))
14184, 140ralsn 4686 . . . . . . . . . . . . . . . . 17 (∀𝑡 ∈ {𝑎}∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On ↔ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
142137, 141sylibr 233 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑡 ∈ {𝑎}∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On)
143 snssi 4812 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → {𝑎} ⊆ On)
144 onss 7772 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ On → 𝑏 ⊆ On)
145 xpss12 5692 . . . . . . . . . . . . . . . . . . . 20 (({𝑎} ⊆ On ∧ 𝑏 ⊆ On) → ({𝑎} × 𝑏) ⊆ (On × On))
146143, 144, 145syl2an 597 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ({𝑎} × 𝑏) ⊆ (On × On))
147146adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ({𝑎} × 𝑏) ⊆ (On × On))
14881fndmi 6654 . . . . . . . . . . . . . . . . . 18 dom +no = (On × On)
149147, 148sseqtrrdi 4034 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ({𝑎} × 𝑏) ⊆ dom +no )
150 funimassov 7584 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ ({𝑎} × 𝑏) ⊆ dom +no ) → (( +no “ ({𝑎} × 𝑏)) ⊆ On ↔ ∀𝑡 ∈ {𝑎}∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On))
15183, 149, 150sylancr 588 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ ({𝑎} × 𝑏)) ⊆ On ↔ ∀𝑡 ∈ {𝑎}∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On))
152142, 151mpbird 257 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ ({𝑎} × 𝑏)) ⊆ On)
153 simprl 770 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On)
154 oveq2 7417 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (𝑐 +no 𝑡) = (𝑐 +no 𝑏))
155154eleq1d 2819 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → ((𝑐 +no 𝑡) ∈ On ↔ (𝑐 +no 𝑏) ∈ On))
15686, 155ralsn 4686 . . . . . . . . . . . . . . . . . 18 (∀𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On ↔ (𝑐 +no 𝑏) ∈ On)
157156ralbii 3094 . . . . . . . . . . . . . . . . 17 (∀𝑐𝑎𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On ↔ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On)
158153, 157sylibr 233 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑐𝑎𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On)
159 onss 7772 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → 𝑎 ⊆ On)
160 snssi 4812 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ On → {𝑏} ⊆ On)
161 xpss12 5692 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ⊆ On ∧ {𝑏} ⊆ On) → (𝑎 × {𝑏}) ⊆ (On × On))
162159, 160, 161syl2an 597 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 × {𝑏}) ⊆ (On × On))
163162adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 × {𝑏}) ⊆ (On × On))
164163, 148sseqtrrdi 4034 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 × {𝑏}) ⊆ dom +no )
165 funimassov 7584 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ (𝑎 × {𝑏}) ⊆ dom +no ) → (( +no “ (𝑎 × {𝑏})) ⊆ On ↔ ∀𝑐𝑎𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On))
16683, 164, 165sylancr 588 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ (𝑎 × {𝑏})) ⊆ On ↔ ∀𝑐𝑎𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On))
167158, 166mpbird 257 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ (𝑎 × {𝑏})) ⊆ On)
168152, 167unssd 4187 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ On)
169 ssorduni 7766 . . . . . . . . . . . . . 14 ((( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ On → Ord (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
170168, 169syl 17 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → Ord (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
171 vsnex 5430 . . . . . . . . . . . . . . . . . 18 {𝑎} ∈ V
172171, 86xpex 7740 . . . . . . . . . . . . . . . . 17 ({𝑎} × 𝑏) ∈ V
173 funimaexg 6635 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ ({𝑎} × 𝑏) ∈ V) → ( +no “ ({𝑎} × 𝑏)) ∈ V)
17483, 172, 173mp2an 691 . . . . . . . . . . . . . . . 16 ( +no “ ({𝑎} × 𝑏)) ∈ V
175 vsnex 5430 . . . . . . . . . . . . . . . . . 18 {𝑏} ∈ V
17684, 175xpex 7740 . . . . . . . . . . . . . . . . 17 (𝑎 × {𝑏}) ∈ V
177 funimaexg 6635 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ (𝑎 × {𝑏}) ∈ V) → ( +no “ (𝑎 × {𝑏})) ∈ V)
17883, 176, 177mp2an 691 . . . . . . . . . . . . . . . 16 ( +no “ (𝑎 × {𝑏})) ∈ V
179174, 178unex 7733 . . . . . . . . . . . . . . 15 (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ V
180179uniex 7731 . . . . . . . . . . . . . 14 (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ V
181180elon 6374 . . . . . . . . . . . . 13 ( (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On ↔ Ord (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
182170, 181sylibr 233 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On)
183 onsucb 7805 . . . . . . . . . . . 12 ( (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On ↔ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On)
184182, 183sylib 217 . . . . . . . . . . 11 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On)
185 onsucuni 7816 . . . . . . . . . . . . 13 ((( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ On → (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
186168, 185syl 17 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
187186unssad 4188 . . . . . . . . . . 11 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ ({𝑎} × 𝑏)) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
188186unssbd 4189 . . . . . . . . . . 11 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ (𝑎 × {𝑏})) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))
189 sseq2 4009 . . . . . . . . . . . . 13 (𝑥 = suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) → (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑎} × 𝑏)) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏})))))
190 sseq2 4009 . . . . . . . . . . . . 13 (𝑥 = suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) → (( +no “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑎 × {𝑏})) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏})))))
191189, 190anbi12d 632 . . . . . . . . . . . 12 (𝑥 = suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) → ((( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑎} × 𝑏)) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∧ ( +no “ (𝑎 × {𝑏})) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))))
192191rspcev 3613 . . . . . . . . . . 11 ((suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ On ∧ (( +no “ ({𝑎} × 𝑏)) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∧ ( +no “ (𝑎 × {𝑏})) ⊆ suc (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))))) → ∃𝑥 ∈ On (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥))
193184, 187, 188, 192syl12anc 836 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∃𝑥 ∈ On (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥))
194 onintrab2 7785 . . . . . . . . . 10 (∃𝑥 ∈ On (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} ∈ On)
195193, 194sylib 217 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} ∈ On)
196136, 195eqeltrd 2834 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)} ∈ On)
19784, 86op1std 7985 . . . . . . . . . . . . . . . 16 (𝑡 = ⟨𝑎, 𝑏⟩ → (1st𝑡) = 𝑎)
198197sneqd 4641 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → {(1st𝑡)} = {𝑎})
19984, 86op2ndd 7986 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → (2nd𝑡) = 𝑏)
200198, 199xpeq12d 5708 . . . . . . . . . . . . . 14 (𝑡 = ⟨𝑎, 𝑏⟩ → ({(1st𝑡)} × (2nd𝑡)) = ({𝑎} × 𝑏))
201200imaeq2d 6060 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑎, 𝑏⟩ → (𝑓 “ ({(1st𝑡)} × (2nd𝑡))) = (𝑓 “ ({𝑎} × 𝑏)))
202201sseq1d 4014 . . . . . . . . . . . 12 (𝑡 = ⟨𝑎, 𝑏⟩ → ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ↔ (𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥))
203199sneqd 4641 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → {(2nd𝑡)} = {𝑏})
204197, 203xpeq12d 5708 . . . . . . . . . . . . . 14 (𝑡 = ⟨𝑎, 𝑏⟩ → ((1st𝑡) × {(2nd𝑡)}) = (𝑎 × {𝑏}))
205204imaeq2d 6060 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑎, 𝑏⟩ → (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) = (𝑓 “ (𝑎 × {𝑏})))
206205sseq1d 4014 . . . . . . . . . . . 12 (𝑡 = ⟨𝑎, 𝑏⟩ → ((𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥 ↔ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥))
207202, 206anbi12d 632 . . . . . . . . . . 11 (𝑡 = ⟨𝑎, 𝑏⟩ → (((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥) ↔ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)))
208207rabbidv 3441 . . . . . . . . . 10 (𝑡 = ⟨𝑎, 𝑏⟩ → {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)})
209208inteqd 4956 . . . . . . . . 9 (𝑡 = ⟨𝑎, 𝑏⟩ → {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)})
210 imaeq1 6055 . . . . . . . . . . . . 13 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → (𝑓 “ ({𝑎} × 𝑏)) = (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)))
211210sseq1d 4014 . . . . . . . . . . . 12 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥))
212 imaeq1 6055 . . . . . . . . . . . . 13 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → (𝑓 “ (𝑎 × {𝑏})) = (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})))
213212sseq1d 4014 . . . . . . . . . . . 12 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → ((𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥))
214211, 213anbi12d 632 . . . . . . . . . . 11 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → (((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)))
215214rabbidv 3441 . . . . . . . . . 10 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → {𝑥 ∈ On ∣ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)})
216215inteqd 4956 . . . . . . . . 9 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → {𝑥 ∈ On ∣ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)})
217 eqid 2733 . . . . . . . . 9 (𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)}) = (𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)})
218209, 216, 217ovmpog 7567 . . . . . . . 8 ((⟨𝑎, 𝑏⟩ ∈ V ∧ ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) ∈ V ∧ {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)} ∈ On) → (⟨𝑎, 𝑏⟩(𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)})( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))) = {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)})
21980, 91, 196, 218mp3an12i 1466 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (⟨𝑎, 𝑏⟩(𝑡 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ On ∣ ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥)})( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}))) = {𝑥 ∈ On ∣ ((( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})) ⊆ 𝑥)})
22079, 219, 1363eqtrd 2777 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)})
221220, 195eqeltrd 2834 . . . . 5 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 +no 𝑏) ∈ On)
222221, 220jca 513 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ((𝑎 +no 𝑏) ∈ On ∧ (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)}))
223222ex 414 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On) → ((𝑎 +no 𝑏) ∈ On ∧ (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)})))
22476, 223syl5 34 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 ((𝑐 +no 𝑑) ∈ On ∧ (𝑐 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)}) ∧ ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) ∧ ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)})) → ((𝑎 +no 𝑏) ∈ On ∧ (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)})))
22514, 28, 41, 55, 69, 224on2ind 8668 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cdif 3946  cun 3947  wss 3949  {csn 4629  cop 4635   cuni 4909   cint 4951   × cxp 5675  dom cdm 5677  cres 5679  cima 5680  Ord word 6364  Oncon0 6365  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  cfv 6544  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974   +no cnadd 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-nadd 8665
This theorem is referenced by:  naddcl  8676  naddov  8677
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