Theorem elsucg 5792

Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)

Assertion

Ref	Expression
elsucg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Proof of Theorem elsucg

Step	Hyp	Ref	Expression
1		df-suc 5729	. . . 4 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵})
2	1	eleq2i 2693	. . 3 ⊢ (𝐴 ∈ suc 𝐵 ↔ 𝐴 ∈ (𝐵 ∪ {𝐵}))
3		elun 3753	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
4	2, 3	bitri 264	. 2 ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}))
5		elsng 4191	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
6	5	orbi2d 738	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ {𝐵}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	4, 6	syl5bb 272	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ∪ cun 3572  {csn 4177  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729

This theorem is referenced by:  elsuc  5794  elelsuc  5797  sucidg  5803  ordsssuc  5812  ordsucelsuc  7022  suc11reg  8516  nlt1pi  9728