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Theorem elsuc2g 5757
Description: Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsuc2g (𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Proof of Theorem elsuc2g
StepHypRef Expression
1 df-suc 5693 . . 3 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2690 . 2 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3736 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
4 elsn2g 4186 . . . 4 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
54orbi2d 737 . . 3 (𝐵𝑉 → ((𝐴𝐵𝐴 ∈ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
63, 5syl5bb 272 . 2 (𝐵𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
72, 6syl5bb 272 1 (𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383   = wceq 1480  wcel 1987  cun 3557  {csn 4153  suc csuc 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564  df-sn 4154  df-suc 5693
This theorem is referenced by:  elsuc2  5759  om2uzlti  12696
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