Theorem limsuc2 43030

Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Assertion

Ref	Expression
limsuc2	⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))

Proof of Theorem limsuc2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordunisuc2 7820	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2	1	biimpa 476	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
3		suceq 6400	. . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspccva 3587	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
6	2, 5	sylan 580	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
7	6	ex 412	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
8		ordtr 6346	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
9		trsuc 6421	. . . . 5 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
10	9	ex 412	. . . 4 ⊢ (Tr 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
11	8, 10	syl 17	. . 3 ⊢ (Ord 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
12	11	adantr 480	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
13	7, 12	impbid 212	1 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∪ cuni 4871  Tr wtr 5214  Ord word 6331  suc csuc 6334

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-suc 6338

This theorem is referenced by:  aomclem4  43046