Theorem ordsucun 7542

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 6294	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
2		ordsuc 7531	. . . . 5 ⊢ (Ord (𝐴 ∪ 𝐵) ↔ Ord suc (𝐴 ∪ 𝐵))
3		ordelon 6217	. . . . . 6 ⊢ ((Ord suc (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ suc (𝐴 ∪ 𝐵)) → 𝑥 ∈ On)
4	3	ex 415	. . . . 5 ⊢ (Ord suc (𝐴 ∪ 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
5	2, 4	sylbi 219	. . . 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 7531	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
8		ordsuc 7531	. . . 4 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
9		ordun 6294	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10		ordelon 6217	. . . . . 6 ⊢ ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
11	10	ex 415	. . . . 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 599	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14		ordssun 6292	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
15	14	adantl 484	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
16		ordsssuc 6279	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ Ord (𝐴 ∪ 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
17	1, 16	sylan2 594	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
18		ordsssuc 6279	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
19	18	adantrr 715	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴))
20		ordsssuc 6279	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
21	20	adantrl 714	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
22	19, 21	orbi12d 915	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
23	15, 17, 22	3bitr3d 311	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
24		elun 4127	. . . . 5 ⊢ (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵))
25	23, 24	syl6bbr 291	. . . 4 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
26	25	expcom 416	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
27	6, 13, 26	pm5.21ndd 383	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
28	27	eqrdv 2821	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ⊆ wss 3938  Ord word 6192  Oncon0 6193  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-suc 6199

This theorem is referenced by:  rankprb  9282  noetalem4  33222