Theorem ordsucun 7025

Description: The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)

Assertion

Ref	Expression
ordsucun	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))

Proof of Theorem ordsucun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordun 5829	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
2		ordsuc 7014	. . . . 5 ⊢ (Ord (𝐴 ∪ 𝐵) ↔ Ord suc (𝐴 ∪ 𝐵))
3		ordelon 5747	. . . . . 6 ⊢ ((Ord suc (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ suc (𝐴 ∪ 𝐵)) → 𝑥 ∈ On)
4	3	ex 450	. . . . 5 ⊢ (Ord suc (𝐴 ∪ 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
5	2, 4	sylbi 207	. . . 4 ⊢ (Ord (𝐴 ∪ 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
6	1, 5	syl 17	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
7		ordsuc 7014	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
8		ordsuc 7014	. . . 4 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
9		ordun 5829	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10		ordelon 5747	. . . . . 6 ⊢ ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
11	10	ex 450	. . . . 5 ⊢ (Ord (suc 𝐴 ∪ suc 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
12	9, 11	syl 17	. . . 4 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
13	7, 8, 12	syl2anb 496	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14		ordssun 5827	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
15	14	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
16		ordsssuc 5812	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ Ord (𝐴 ∪ 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
17	1, 16	sylan2 491	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
18		ordsssuc 5812	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
19	18	adantrr 753	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
20		ordsssuc 5812	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
21	20	adantrl 752	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
22	19, 21	orbi12d 746	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
23	15, 17, 22	3bitr3d 298	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
24		elun 3753	. . . . 5 ⊢ (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵))
25	23, 24	syl6bbr 278	. . . 4 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
26	25	expcom 451	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
27	6, 13, 26	pm5.21ndd 369	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
28	27	eqrdv 2620	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  Ord word 5722  Oncon0 5723  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  rankprb  8714  noetalem4  31866