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Theorem naddgeoa 42721
Description: Natural addition results in a value greater than or equal than that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
Assertion
Ref Expression
naddgeoa ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))

Proof of Theorem naddgeoa
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7412 . . 3 (𝑎 = 𝑐 → (𝑎 +o 𝑏) = (𝑐 +o 𝑏))
2 oveq1 7412 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
31, 2sseq12d 4010 . 2 (𝑎 = 𝑐 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏)))
4 oveq2 7413 . . 3 (𝑏 = 𝑑 → (𝑐 +o 𝑏) = (𝑐 +o 𝑑))
5 oveq2 7413 . . 3 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
64, 5sseq12d 4010 . 2 (𝑏 = 𝑑 → ((𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
7 oveq1 7412 . . 3 (𝑎 = 𝑐 → (𝑎 +o 𝑑) = (𝑐 +o 𝑑))
8 oveq1 7412 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
97, 8sseq12d 4010 . 2 (𝑎 = 𝑐 → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
10 oveq1 7412 . . 3 (𝑎 = 𝐴 → (𝑎 +o 𝑏) = (𝐴 +o 𝑏))
11 oveq1 7412 . . 3 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
1210, 11sseq12d 4010 . 2 (𝑎 = 𝐴 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏)))
13 oveq2 7413 . . 3 (𝑏 = 𝐵 → (𝐴 +o 𝑏) = (𝐴 +o 𝐵))
14 oveq2 7413 . . 3 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
1513, 14sseq12d 4010 . 2 (𝑏 = 𝐵 → ((𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏) ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵)))
16 simplll 772 . . . . . 6 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑎 ∈ On)
17 simpllr 773 . . . . . . 7 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑏 ∈ On)
18 simplr 766 . . . . . . 7 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → Lim 𝑏)
1917, 18jca 511 . . . . . 6 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑏 ∈ On ∧ Lim 𝑏))
20 oalim 8533 . . . . . 6 ((𝑎 ∈ On ∧ (𝑏 ∈ On ∧ Lim 𝑏)) → (𝑎 +o 𝑏) = 𝑑𝑏 (𝑎 +o 𝑑))
2116, 19, 20syl2anc 583 . . . . 5 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = 𝑑𝑏 (𝑎 +o 𝑑))
22 simpl 482 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) → (𝑎 ∈ On ∧ 𝑏 ∈ On))
23 simp3 1135 . . . . . 6 ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
24 simpr 484 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
25 simpr 484 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → 𝑏 ∈ On)
26 onelss 6400 . . . . . . . . . . . . . . 15 (𝑏 ∈ On → (𝑑𝑏𝑑𝑏))
2725, 26syl 17 . . . . . . . . . . . . . 14 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑑𝑏𝑑𝑏))
2827imp 406 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → 𝑑𝑏)
29 simplr 766 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → 𝑏 ∈ On)
30 simpr 484 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → 𝑑𝑏)
31 onelon 6383 . . . . . . . . . . . . . . 15 ((𝑏 ∈ On ∧ 𝑑𝑏) → 𝑑 ∈ On)
3229, 30, 31syl2anc 583 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → 𝑑 ∈ On)
33 simpll 764 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → 𝑎 ∈ On)
34 naddss2 8691 . . . . . . . . . . . . . 14 ((𝑑 ∈ On ∧ 𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑑𝑏 ↔ (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏)))
3532, 29, 33, 34syl3anc 1368 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → (𝑑𝑏 ↔ (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏)))
3628, 35mpbid 231 . . . . . . . . . . . 12 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏))
3736adantr 480 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏))
3824, 37sstrd 3987 . . . . . . . . . 10 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
3938ex 412 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
4039ralimdva 3161 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
4140imp 406 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
42 iunss 5041 . . . . . . 7 ( 𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏) ↔ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
4341, 42sylibr 233 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → 𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
4422, 23, 43syl2an 595 . . . . 5 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
4521, 44eqsstrd 4015 . . . 4 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
4645exp31 419 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (Lim 𝑏 → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
47 dflim3 7833 . . . . . . 7 (Lim 𝑏 ↔ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
4847notbii 320 . . . . . 6 (¬ Lim 𝑏 ↔ ¬ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
49 iman 401 . . . . . 6 ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ ¬ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
5048, 49bitr4i 278 . . . . 5 (¬ Lim 𝑏 ↔ (Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
51 eloni 6368 . . . . . 6 (𝑏 ∈ On → Ord 𝑏)
52 pm5.5 361 . . . . . 6 (Ord 𝑏 → ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
5325, 51, 523syl 18 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
5450, 53bitrid 283 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (¬ Lim 𝑏 ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
55 ssidd 4000 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎𝑎)
56 simpr 484 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑏 = ∅)
5756oveq2d 7421 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = (𝑎 +o ∅))
58 simpll 764 . . . . . . . . . 10 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎 ∈ On)
59 oa0 8517 . . . . . . . . . 10 (𝑎 ∈ On → (𝑎 +o ∅) = 𝑎)
6058, 59syl 17 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o ∅) = 𝑎)
6157, 60eqtrd 2766 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = 𝑎)
6256oveq2d 7421 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = (𝑎 +no ∅))
63 naddrid 8684 . . . . . . . . . 10 (𝑎 ∈ On → (𝑎 +no ∅) = 𝑎)
6458, 63syl 17 . . . . . . . . 9 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no ∅) = 𝑎)
6562, 64eqtrd 2766 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = 𝑎)
6655, 61, 653sstr4d 4024 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
6766a1d 25 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
6867ex 412 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 = ∅ → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
69 vex 3472 . . . . . . . . . . 11 𝑑 ∈ V
7069sucid 6440 . . . . . . . . . 10 𝑑 ∈ suc 𝑑
71 simpr 484 . . . . . . . . . 10 ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑏 = suc 𝑑)
7270, 71eleqtrrid 2834 . . . . . . . . 9 ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑑𝑏)
7372, 71jca 511 . . . . . . . 8 ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → (𝑑𝑏𝑏 = suc 𝑑))
7473a1i 11 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → (𝑑𝑏𝑏 = suc 𝑑)))
7574reximdv2 3158 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ On 𝑏 = suc 𝑑 → ∃𝑑𝑏 𝑏 = suc 𝑑))
76 r19.29r 3110 . . . . . . . . 9 ((∃𝑑𝑏 𝑏 = suc 𝑑 ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∃𝑑𝑏 (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)))
77 simprr 770 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
7833, 32jca 511 . . . . . . . . . . . . . 14 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
79 oacl 8536 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +o 𝑑) ∈ On)
80 eloni 6368 . . . . . . . . . . . . . . . 16 ((𝑎 +o 𝑑) ∈ On → Ord (𝑎 +o 𝑑))
8179, 80syl 17 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → Ord (𝑎 +o 𝑑))
82 naddcl 8678 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no 𝑑) ∈ On)
83 eloni 6368 . . . . . . . . . . . . . . . 16 ((𝑎 +no 𝑑) ∈ On → Ord (𝑎 +no 𝑑))
8482, 83syl 17 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → Ord (𝑎 +no 𝑑))
8581, 84jca 511 . . . . . . . . . . . . . 14 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (Ord (𝑎 +o 𝑑) ∧ Ord (𝑎 +no 𝑑)))
86 ordsucsssuc 7808 . . . . . . . . . . . . . 14 ((Ord (𝑎 +o 𝑑) ∧ Ord (𝑎 +no 𝑑)) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
8778, 85, 863syl 18 . . . . . . . . . . . . 13 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
8887adantr 480 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
8977, 88mpbid 231 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑))
90 simprl 768 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑏 = suc 𝑑)
9190oveq2d 7421 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = (𝑎 +o suc 𝑑))
9278adantr 480 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
93 oasuc 8525 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +o suc 𝑑) = suc (𝑎 +o 𝑑))
9492, 93syl 17 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o suc 𝑑) = suc (𝑎 +o 𝑑))
9591, 94eqtrd 2766 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = suc (𝑎 +o 𝑑))
9690oveq2d 7421 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no 𝑏) = (𝑎 +no suc 𝑑))
97 simplll 772 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑎 ∈ On)
9831ad4ant23 750 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑑 ∈ On)
99 naddsuc2 42719 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no suc 𝑑) = suc (𝑎 +no 𝑑))
10097, 98, 99syl2anc 583 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no suc 𝑑) = suc (𝑎 +no 𝑑))
10196, 100eqtrd 2766 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no 𝑏) = suc (𝑎 +no 𝑑))
10289, 95, 1013sstr4d 4024 . . . . . . . . . 10 ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
103102rexlimdva2 3151 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑𝑏 (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
10476, 103syl5 34 . . . . . . . 8 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∃𝑑𝑏 𝑏 = suc 𝑑 ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
105104expd 415 . . . . . . 7 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑𝑏 𝑏 = suc 𝑑 → (∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
10623, 105syl7 74 . . . . . 6 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑𝑏 𝑏 = suc 𝑑 → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
10775, 106syld 47 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ On 𝑏 = suc 𝑑 → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
10868, 107jaod 856 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑) → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
10954, 108sylbid 239 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (¬ Lim 𝑏 → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
11046, 109pm2.61d 179 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
1113, 6, 9, 12, 15, 110on2ind 8670 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  c0 4317   ciun 4990  Ord word 6357  Oncon0 6358  Lim wlim 6359  suc csuc 6360  (class class class)co 7405   +o coa 8464   +no cnadd 8666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-oadd 8471  df-nadd 8667
This theorem is referenced by:  naddwordnexlem4  42728
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