Theorem limsuc2 39661

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 7559	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2	1	biimpa 479	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
3		suceq 6256	. . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2897	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspccva 3622	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
6	2, 5	sylan 582	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
7	6	ex 415	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
8		ordtr 6205	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
9		trsuc 6275	. . . . 5 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
10	9	ex 415	. . . 4 ⊢ (Tr 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
11	8, 10	syl 17	. . 3 ⊢ (Ord 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
12	11	adantr 483	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
13	7, 12	impbid 214	1 ⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∪ cuni 4838  Tr wtr 5172  Ord word 6190  suc csuc 6193

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 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-ord 6194  df-on 6195  df-suc 6197

This theorem is referenced by:  aomclem4  39677