Theorem ordsucun 7845

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 6488	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
2		ordsuc 7833	. . . . 5 ⊢ (Ord (𝐴 ∪ 𝐵) ↔ Ord suc (𝐴 ∪ 𝐵))
3		ordelon 6408	. . . . . 6 ⊢ ((Ord suc (𝐴 ∪ 𝐵) ∧ 𝑥 ∈ suc (𝐴 ∪ 𝐵)) → 𝑥 ∈ On)
4	3	ex 412	. . . . 5 ⊢ (Ord suc (𝐴 ∪ 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) → 𝑥 ∈ On))
5	2, 4	sylbi 217	. . . 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 7833	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
8		ordsuc 7833	. . . 4 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
9		ordun 6488	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord suc 𝐵) → Ord (suc 𝐴 ∪ suc 𝐵))
10		ordelon 6408	. . . . . 6 ⊢ ((Ord (suc 𝐴 ∪ suc 𝐵) ∧ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)) → 𝑥 ∈ On)
11	10	ex 412	. . . . 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 598	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) → 𝑥 ∈ On))
14		ordssun 6486	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
15	14	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵)))
16		ordsssuc 6473	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ Ord (𝐴 ∪ 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
17	1, 16	sylan2 593	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
18		ordsssuc 6473	. . . . . . . 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 6473	. . . . . . . 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 309	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵)))
24		elun 4153	. . . . 5 ⊢ (𝑥 ∈ (suc 𝐴 ∪ suc 𝐵) ↔ (𝑥 ∈ suc 𝐴 ∨ 𝑥 ∈ suc 𝐵))
25	23, 24	bitr4di 289	. . . 4 ⊢ ((𝑥 ∈ On ∧ (Ord 𝐴 ∧ Ord 𝐵)) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
26	25	expcom 413	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ On → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵))))
27	6, 13, 26	pm5.21ndd 379	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ suc (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (suc 𝐴 ∪ suc 𝐵)))
28	27	eqrdv 2735	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴 ∪ 𝐵) = (suc 𝐴 ∪ suc 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ∪ cun 3949   ⊆ wss 3951  Ord word 6383  Oncon0 6384  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by:  rankprb  9891