Theorem limsuc2 43074

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 7769	. . . . 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 6369	. . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2816	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspccva 3571	. . . 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 6315	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
9		trsuc 6390	. . . . 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 1541   ∈ wcel 2111  ∀wral 3047  ∪ cuni 4854  Tr wtr 5193  Ord word 6300  suc csuc 6303

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-ord 6304  df-on 6305  df-suc 6307

This theorem is referenced by:  aomclem4  43090