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Theorem elsuc2g 6281
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 6219 . . 3 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2829 . 2 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 4063 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
4 elsn2g 4579 . . . 4 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
54orbi2d 916 . . 3 (𝐵𝑉 → ((𝐴𝐵𝐴 ∈ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
63, 5syl5bb 286 . 2 (𝐵𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
72, 6syl5bb 286 1 (𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wo 847   = wceq 1543  wcel 2110  cun 3864  {csn 4541  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-un 3871  df-sn 4542  df-suc 6219
This theorem is referenced by:  elsuc2  6283  om2uzlti  13523
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