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Theorem naddcllem 8677
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 7418 . . . 4 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
21eleq1d 2818 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) ∈ On ↔ (𝑐 +no 𝑏) ∈ On))
3 sneq 4638 . . . . . . . . . 10 (𝑎 = 𝑐 → {𝑎} = {𝑐})
43xpeq1d 5705 . . . . . . . . 9 (𝑎 = 𝑐 → ({𝑎} × 𝑏) = ({𝑐} × 𝑏))
54imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ ({𝑎} × 𝑏)) = ( +no “ ({𝑐} × 𝑏)))
65sseq1d 4013 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥))
7 xpeq1 5690 . . . . . . . . 9 (𝑎 = 𝑐 → (𝑎 × {𝑏}) = (𝑐 × {𝑏}))
87imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ (𝑎 × {𝑏})) = ( +no “ (𝑐 × {𝑏})))
98sseq1d 4013 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥))
106, 9anbi12d 631 . . . . . 6 (𝑎 = 𝑐 → ((( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)))
1110rabbidv 3440 . . . . 5 (𝑎 = 𝑐 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)})
1211inteqd 4955 . . . 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 2748 . . 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 631 . 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 7419 . . . 4 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
1615eleq1d 2818 . . 3 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) ∈ On ↔ (𝑐 +no 𝑑) ∈ On))
17 xpeq2 5697 . . . . . . . . 9 (𝑏 = 𝑑 → ({𝑐} × 𝑏) = ({𝑐} × 𝑑))
1817imaeq2d 6059 . . . . . . . 8 (𝑏 = 𝑑 → ( +no “ ({𝑐} × 𝑏)) = ( +no “ ({𝑐} × 𝑑)))
1918sseq1d 4013 . . . . . . 7 (𝑏 = 𝑑 → (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥))
20 sneq 4638 . . . . . . . . . 10 (𝑏 = 𝑑 → {𝑏} = {𝑑})
2120xpeq2d 5706 . . . . . . . . 9 (𝑏 = 𝑑 → (𝑐 × {𝑏}) = (𝑐 × {𝑑}))
2221imaeq2d 6059 . . . . . . . 8 (𝑏 = 𝑑 → ( +no “ (𝑐 × {𝑏})) = ( +no “ (𝑐 × {𝑑})))
2322sseq1d 4013 . . . . . . 7 (𝑏 = 𝑑 → (( +no “ (𝑐 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥))
2419, 23anbi12d 631 . . . . . 6 (𝑏 = 𝑑 → ((( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)))
2524rabbidv 3440 . . . . 5 (𝑏 = 𝑑 → {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
2625inteqd 4955 . . . 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 2748 . . 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 631 . 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 7418 . . . 4 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
3029eleq1d 2818 . . 3 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) ∈ On ↔ (𝑐 +no 𝑑) ∈ On))
313xpeq1d 5705 . . . . . . . . 9 (𝑎 = 𝑐 → ({𝑎} × 𝑑) = ({𝑐} × 𝑑))
3231imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ ({𝑎} × 𝑑)) = ( +no “ ({𝑐} × 𝑑)))
3332sseq1d 4013 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ↔ ( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥))
34 xpeq1 5690 . . . . . . . . 9 (𝑎 = 𝑐 → (𝑎 × {𝑑}) = (𝑐 × {𝑑}))
3534imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝑐 → ( +no “ (𝑎 × {𝑑})) = ( +no “ (𝑐 × {𝑑})))
3635sseq1d 4013 . . . . . . 7 (𝑎 = 𝑐 → (( +no “ (𝑎 × {𝑑})) ⊆ 𝑥 ↔ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥))
3733, 36anbi12d 631 . . . . . 6 (𝑎 = 𝑐 → ((( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥) ↔ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)))
3837rabbidv 3440 . . . . 5 (𝑎 = 𝑐 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑑})) ⊆ 𝑥)})
3938inteqd 4955 . . . 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 2748 . . 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 631 . 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 7418 . . . 4 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
4342eleq1d 2818 . . 3 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) ∈ On ↔ (𝐴 +no 𝑏) ∈ On))
44 sneq 4638 . . . . . . . . . 10 (𝑎 = 𝐴 → {𝑎} = {𝐴})
4544xpeq1d 5705 . . . . . . . . 9 (𝑎 = 𝐴 → ({𝑎} × 𝑏) = ({𝐴} × 𝑏))
4645imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝐴 → ( +no “ ({𝑎} × 𝑏)) = ( +no “ ({𝐴} × 𝑏)))
4746sseq1d 4013 . . . . . . 7 (𝑎 = 𝐴 → (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥))
48 xpeq1 5690 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎 × {𝑏}) = (𝐴 × {𝑏}))
4948imaeq2d 6059 . . . . . . . 8 (𝑎 = 𝐴 → ( +no “ (𝑎 × {𝑏})) = ( +no “ (𝐴 × {𝑏})))
5049sseq1d 4013 . . . . . . 7 (𝑎 = 𝐴 → (( +no “ (𝑎 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥))
5147, 50anbi12d 631 . . . . . 6 (𝑎 = 𝐴 → ((( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)))
5251rabbidv 3440 . . . . 5 (𝑎 = 𝐴 → {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)})
5352inteqd 4955 . . . 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 2748 . . 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 631 . 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 7419 . . . 4 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
5756eleq1d 2818 . . 3 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) ∈ On ↔ (𝐴 +no 𝐵) ∈ On))
58 xpeq2 5697 . . . . . . . . 9 (𝑏 = 𝐵 → ({𝐴} × 𝑏) = ({𝐴} × 𝐵))
5958imaeq2d 6059 . . . . . . . 8 (𝑏 = 𝐵 → ( +no “ ({𝐴} × 𝑏)) = ( +no “ ({𝐴} × 𝐵)))
6059sseq1d 4013 . . . . . . 7 (𝑏 = 𝐵 → (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ↔ ( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥))
61 sneq 4638 . . . . . . . . . 10 (𝑏 = 𝐵 → {𝑏} = {𝐵})
6261xpeq2d 5706 . . . . . . . . 9 (𝑏 = 𝐵 → (𝐴 × {𝑏}) = (𝐴 × {𝐵}))
6362imaeq2d 6059 . . . . . . . 8 (𝑏 = 𝐵 → ( +no “ (𝐴 × {𝑏})) = ( +no “ (𝐴 × {𝐵})))
6463sseq1d 4013 . . . . . . 7 (𝑏 = 𝐵 → (( +no “ (𝐴 × {𝑏})) ⊆ 𝑥 ↔ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥))
6560, 64anbi12d 631 . . . . . 6 (𝑏 = 𝐵 → ((( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)))
6665rabbidv 3440 . . . . 5 (𝑏 = 𝐵 → {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝑏})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
6766inteqd 4955 . . . 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 2748 . . 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 631 . 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 483 . . . . . 6 (((𝑐 +no 𝑏) ∈ On ∧ (𝑐 +no 𝑏) = {𝑥 ∈ On ∣ (( +no “ ({𝑐} × 𝑏)) ⊆ 𝑥 ∧ ( +no “ (𝑐 × {𝑏})) ⊆ 𝑥)}) → (𝑐 +no 𝑏) ∈ On)
7170ralimi 3083 . . . . 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 1134 . . . 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 483 . . . . . 6 (((𝑎 +no 𝑑) ∈ On ∧ (𝑎 +no 𝑑) = {𝑥 ∈ On ∣ (( +no “ ({𝑎} × 𝑑)) ⊆ 𝑥 ∧ ( +no “ (𝑎 × {𝑑})) ⊆ 𝑥)}) → (𝑎 +no 𝑑) ∈ On)
7473ralimi 3083 . . . . 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 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)})) → ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
7672, 75jca 512 . . 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 8667 . . . . . . . . 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 8669 . . . . . . . 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 481 . . . . . . 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 5464 . . . . . . . 8 𝑎, 𝑏⟩ ∈ V
81 naddfn 8676 . . . . . . . . . 10 +no Fn (On × On)
82 fnfun 6649 . . . . . . . . . 10 ( +no Fn (On × On) → Fun +no )
8381, 82ax-mp 5 . . . . . . . . 9 Fun +no
84 vex 3478 . . . . . . . . . . . 12 𝑎 ∈ V
8584sucex 7796 . . . . . . . . . . 11 suc 𝑎 ∈ V
86 vex 3478 . . . . . . . . . . . 12 𝑏 ∈ V
8786sucex 7796 . . . . . . . . . . 11 suc 𝑏 ∈ V
8885, 87xpex 7742 . . . . . . . . . 10 (suc 𝑎 × suc 𝑏) ∈ V
8988difexi 5328 . . . . . . . . 9 ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ∈ V
90 resfunexg 7219 . . . . . . . . 9 ((Fun +no ∧ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ∈ V) → ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) ∈ V)
9183, 89, 90mp2an 690 . . . . . . . 8 ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) ∈ V
92 eloni 6374 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ On → Ord 𝑏)
9392ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → Ord 𝑏)
94 ordirr 6382 . . . . . . . . . . . . . . . . . 18 (Ord 𝑏 → ¬ 𝑏𝑏)
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ 𝑏𝑏)
9695olcd 872 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (¬ 𝑎 ∈ {𝑎} ∨ ¬ 𝑏𝑏))
97 ianor 980 . . . . . . . . . . . . . . . . 17 (¬ (𝑎 ∈ {𝑎} ∧ 𝑏𝑏) ↔ (¬ 𝑎 ∈ {𝑎} ∨ ¬ 𝑏𝑏))
98 opelxp 5712 . . . . . . . . . . . . . . . . 17 (⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏) ↔ (𝑎 ∈ {𝑎} ∧ 𝑏𝑏))
9997, 98xchnxbir 332 . . . . . . . . . . . . . . . 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 6446 . . . . . . . . . . . . . . . . . 18 𝑎 ∈ suc 𝑎
102 snssi 4811 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ suc 𝑎 → {𝑎} ⊆ suc 𝑎)
103101, 102ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑎} ⊆ suc 𝑎
104 sssucid 6444 . . . . . . . . . . . . . . . . 17 𝑏 ⊆ suc 𝑏
105 xpss12 5691 . . . . . . . . . . . . . . . . 17 (({𝑎} ⊆ suc 𝑎𝑏 ⊆ suc 𝑏) → ({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏))
106103, 104, 105mp2an 690 . . . . . . . . . . . . . . . 16 ({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏)
107 ssdifsn 4791 . . . . . . . . . . . . . . . 16 (({𝑎} × 𝑏) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ (({𝑎} × 𝑏) ⊆ (suc 𝑎 × suc 𝑏) ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ ({𝑎} × 𝑏)))
108106, 107mpbiran 707 . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . 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 4013 . . . . . . . . . . . 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 6374 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ On → Ord 𝑎)
114113ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → Ord 𝑎)
115 ordirr 6382 . . . . . . . . . . . . . . . . . 18 (Ord 𝑎 → ¬ 𝑎𝑎)
116114, 115syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ¬ 𝑎𝑎)
117116orcd 871 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ {𝑏}))
118 ianor 980 . . . . . . . . . . . . . . . . 17 (¬ (𝑎𝑎𝑏 ∈ {𝑏}) ↔ (¬ 𝑎𝑎 ∨ ¬ 𝑏 ∈ {𝑏}))
119 opelxp 5712 . . . . . . . . . . . . . . . . 17 (⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏}) ↔ (𝑎𝑎𝑏 ∈ {𝑏}))
120118, 119xchnxbir 332 . . . . . . . . . . . . . . . 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 6444 . . . . . . . . . . . . . . . . 17 𝑎 ⊆ suc 𝑎
12386sucid 6446 . . . . . . . . . . . . . . . . . 18 𝑏 ∈ suc 𝑏
124 snssi 4811 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ suc 𝑏 → {𝑏} ⊆ suc 𝑏)
125123, 124ax-mp 5 . . . . . . . . . . . . . . . . 17 {𝑏} ⊆ suc 𝑏
126 xpss12 5691 . . . . . . . . . . . . . . . . 17 ((𝑎 ⊆ suc 𝑎 ∧ {𝑏} ⊆ suc 𝑏) → (𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏))
127122, 125, 126mp2an 690 . . . . . . . . . . . . . . . 16 (𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏)
128 ssdifsn 4791 . . . . . . . . . . . . . . . 16 ((𝑎 × {𝑏}) ⊆ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩}) ↔ ((𝑎 × {𝑏}) ⊆ (suc 𝑎 × suc 𝑏) ∧ ¬ ⟨𝑎, 𝑏⟩ ∈ (𝑎 × {𝑏})))
129127, 128mpbiran 707 . . . . . . . . . . . . . . 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 6016 . . . . . . . . . . . . . 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 4013 . . . . . . . . . . . 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 631 . . . . . . . . . . 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 3440 . . . . . . . . . 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 4955 . . . . . . . . 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 771 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)
138 oveq1 7418 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑎 → (𝑡 +no 𝑑) = (𝑎 +no 𝑑))
139138eleq1d 2818 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑎 → ((𝑡 +no 𝑑) ∈ On ↔ (𝑎 +no 𝑑) ∈ On))
140139ralbidv 3177 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑎 → (∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On ↔ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On))
14184, 140ralsn 4685 . . . . . . . . . . . . . . . . 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 4811 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → {𝑎} ⊆ On)
144 onss 7774 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ On → 𝑏 ⊆ On)
145 xpss12 5691 . . . . . . . . . . . . . . . . . . . 20 (({𝑎} ⊆ On ∧ 𝑏 ⊆ On) → ({𝑎} × 𝑏) ⊆ (On × On))
146143, 144, 145syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ({𝑎} × 𝑏) ⊆ (On × On))
147146adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ({𝑎} × 𝑏) ⊆ (On × On))
14881fndmi 6653 . . . . . . . . . . . . . . . . . 18 dom +no = (On × On)
149147, 148sseqtrrdi 4033 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ({𝑎} × 𝑏) ⊆ dom +no )
150 funimassov 7586 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ ({𝑎} × 𝑏) ⊆ dom +no ) → (( +no “ ({𝑎} × 𝑏)) ⊆ On ↔ ∀𝑡 ∈ {𝑎}∀𝑑𝑏 (𝑡 +no 𝑑) ∈ On))
15183, 149, 150sylancr 587 . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ ({𝑎} × 𝑏)) ⊆ On)
153 simprl 769 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On)
154 oveq2 7419 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (𝑐 +no 𝑡) = (𝑐 +no 𝑏))
155154eleq1d 2818 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → ((𝑐 +no 𝑡) ∈ On ↔ (𝑐 +no 𝑏) ∈ On))
15686, 155ralsn 4685 . . . . . . . . . . . . . . . . . 18 (∀𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On ↔ (𝑐 +no 𝑏) ∈ On)
157156ralbii 3093 . . . . . . . . . . . . . . . . 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 7774 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → 𝑎 ⊆ On)
160 snssi 4811 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ On → {𝑏} ⊆ On)
161 xpss12 5691 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ⊆ On ∧ {𝑏} ⊆ On) → (𝑎 × {𝑏}) ⊆ (On × On))
162159, 160, 161syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 × {𝑏}) ⊆ (On × On))
163162adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 × {𝑏}) ⊆ (On × On))
164163, 148sseqtrrdi 4033 . . . . . . . . . . . . . . . . 17 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 × {𝑏}) ⊆ dom +no )
165 funimassov 7586 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ (𝑎 × {𝑏}) ⊆ dom +no ) → (( +no “ (𝑎 × {𝑏})) ⊆ On ↔ ∀𝑐𝑎𝑡 ∈ {𝑏} (𝑐 +no 𝑡) ∈ On))
16683, 164, 165sylancr 587 . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → ( +no “ (𝑎 × {𝑏})) ⊆ On)
168152, 167unssd 4186 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ⊆ On)
169 ssorduni 7768 . . . . . . . . . . . . . 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 5429 . . . . . . . . . . . . . . . . . 18 {𝑎} ∈ V
172171, 86xpex 7742 . . . . . . . . . . . . . . . . 17 ({𝑎} × 𝑏) ∈ V
173 funimaexg 6634 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ ({𝑎} × 𝑏) ∈ V) → ( +no “ ({𝑎} × 𝑏)) ∈ V)
17483, 172, 173mp2an 690 . . . . . . . . . . . . . . . 16 ( +no “ ({𝑎} × 𝑏)) ∈ V
175 vsnex 5429 . . . . . . . . . . . . . . . . . 18 {𝑏} ∈ V
17684, 175xpex 7742 . . . . . . . . . . . . . . . . 17 (𝑎 × {𝑏}) ∈ V
177 funimaexg 6634 . . . . . . . . . . . . . . . . 17 ((Fun +no ∧ (𝑎 × {𝑏}) ∈ V) → ( +no “ (𝑎 × {𝑏})) ∈ V)
17883, 176, 177mp2an 690 . . . . . . . . . . . . . . . 16 ( +no “ (𝑎 × {𝑏})) ∈ V
179174, 178unex 7735 . . . . . . . . . . . . . . 15 (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ V
180179uniex 7733 . . . . . . . . . . . . . 14 (( +no “ ({𝑎} × 𝑏)) ∪ ( +no “ (𝑎 × {𝑏}))) ∈ V
181180elon 6373 . . . . . . . . . . . . 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 7807 . . . . . . . . . . . 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 7818 . . . . . . . . . . . . 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 4187 . . . . . . . . . . 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 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}))))
189 sseq2 4008 . . . . . . . . . . . . 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 4008 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 3612 . . . . . . . . . . 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 835 . . . . . . . . . 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 7787 . . . . . . . . . 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 2833 . . . . . . . 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 7987 . . . . . . . . . . . . . . . 16 (𝑡 = ⟨𝑎, 𝑏⟩ → (1st𝑡) = 𝑎)
198197sneqd 4640 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → {(1st𝑡)} = {𝑎})
19984, 86op2ndd 7988 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → (2nd𝑡) = 𝑏)
200198, 199xpeq12d 5707 . . . . . . . . . . . . . 14 (𝑡 = ⟨𝑎, 𝑏⟩ → ({(1st𝑡)} × (2nd𝑡)) = ({𝑎} × 𝑏))
201200imaeq2d 6059 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑎, 𝑏⟩ → (𝑓 “ ({(1st𝑡)} × (2nd𝑡))) = (𝑓 “ ({𝑎} × 𝑏)))
202201sseq1d 4013 . . . . . . . . . . . 12 (𝑡 = ⟨𝑎, 𝑏⟩ → ((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ↔ (𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥))
203199sneqd 4640 . . . . . . . . . . . . . . 15 (𝑡 = ⟨𝑎, 𝑏⟩ → {(2nd𝑡)} = {𝑏})
204197, 203xpeq12d 5707 . . . . . . . . . . . . . 14 (𝑡 = ⟨𝑎, 𝑏⟩ → ((1st𝑡) × {(2nd𝑡)}) = (𝑎 × {𝑏}))
205204imaeq2d 6059 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑎, 𝑏⟩ → (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) = (𝑓 “ (𝑎 × {𝑏})))
206205sseq1d 4013 . . . . . . . . . . . 12 (𝑡 = ⟨𝑎, 𝑏⟩ → ((𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥 ↔ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥))
207202, 206anbi12d 631 . . . . . . . . . . 11 (𝑡 = ⟨𝑎, 𝑏⟩ → (((𝑓 “ ({(1st𝑡)} × (2nd𝑡))) ⊆ 𝑥 ∧ (𝑓 “ ((1st𝑡) × {(2nd𝑡)})) ⊆ 𝑥) ↔ ((𝑓 “ ({𝑎} × 𝑏)) ⊆ 𝑥 ∧ (𝑓 “ (𝑎 × {𝑏})) ⊆ 𝑥)))
208207rabbidv 3440 . . . . . . . . . 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 4955 . . . . . . . . 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 6054 . . . . . . . . . . . . 13 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → (𝑓 “ ({𝑎} × 𝑏)) = (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ ({𝑎} × 𝑏)))
211210sseq1d 4013 . . . . . . . . . . . 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 6054 . . . . . . . . . . . . 13 (𝑓 = ( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) → (𝑓 “ (𝑎 × {𝑏})) = (( +no ↾ ((suc 𝑎 × suc 𝑏) ∖ {⟨𝑎, 𝑏⟩})) “ (𝑎 × {𝑏})))
213212sseq1d 4013 . . . . . . . . . . . 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 631 . . . . . . . . . . 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 3440 . . . . . . . . . 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 4955 . . . . . . . . 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 2732 . . . . . . . . 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 7569 . . . . . . . 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 1465 . . . . . . 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 2776 . . . . . 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 2833 . . . . 5 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ On ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) ∈ On)) → (𝑎 +no 𝑏) ∈ On)
222221, 220jca 512 . . . 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 413 . . 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 8670 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  cdif 3945  cun 3946  wss 3948  {csn 4628  cop 4634   cuni 4908   cint 4950   × cxp 5674  dom cdm 5676  cres 5678  cima 5679  Ord word 6363  Oncon0 6364  suc csuc 6366  Fun wfun 6537   Fn wfn 6538  cfv 6543  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976   +no cnadd 8666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-nadd 8667
This theorem is referenced by:  naddcl  8678  naddov  8679
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