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Theorem elsuc2g 4223
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 4189 . . 3 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2154 . 2 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3139 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
4 elsn2g 3472 . . . 4 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
54orbi2d 739 . . 3 (𝐵𝑉 → ((𝐴𝐵𝐴 ∈ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
63, 5syl5bb 190 . 2 (𝐵𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
72, 6syl5bb 190 1 (𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664   = wceq 1289  wcel 1438  cun 2995  {csn 3441  suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-suc 4189
This theorem is referenced by:  elsuc2  4225  nntri3or  6236  frec2uzltd  9775
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