Theorem ordsucun 7582

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 6292	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
2		ordsuc 7571	. . . . 5 ⊢ (Ord (𝐴 ∪ 𝐵) ↔ Ord suc (𝐴 ∪ 𝐵))
3		ordelon 6215	. . . . . 6 ⊢ ((Ord suc (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ suc (𝐴 ∪ 𝐵)) → 𝑥 ∈ On)
4	3	ex 416	. . . . 5 ⊢ (Ord suc (𝐴 ∪ 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
5	2, 4	sylbi 220	. . . 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 7571	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
8		ordsuc 7571	. . . 4 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
9		ordun 6292	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10		ordelon 6215	. . . . . 6 ⊢ ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
11	10	ex 416	. . . . 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 601	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14		ordssun 6290	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
15	14	adantl 485	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
16		ordsssuc 6277	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ Ord (𝐴 ∪ 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
17	1, 16	sylan2 596	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
18		ordsssuc 6277	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
19	18	adantrr 717	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
20		ordsssuc 6277	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
21	20	adantrl 716	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
22	19, 21	orbi12d 919	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
23	15, 17, 22	3bitr3d 312	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
24		elun 4049	. . . . 5 ⊢ (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵))
25	23, 24	bitr4di 292	. . . 4 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
26	25	expcom 417	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
27	6, 13, 26	pm5.21ndd 384	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
28	27	eqrdv 2734	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2112   ∪ cun 3851   ⊆ wss 3853  Ord word 6190  Oncon0 6191  suc csuc 6193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-ord 6194  df-on 6195  df-suc 6197

This theorem is referenced by:  rankprb  9432