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Theorem elsuc2g 4169
 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 4135 . . 3 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2120 . 2 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3111 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
4 elsn2g 3431 . . . 4 (𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
54orbi2d 714 . . 3 (𝐵𝑉 → ((𝐴𝐵𝐴 ∈ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
63, 5syl5bb 185 . 2 (𝐵𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 = 𝐵)))
72, 6syl5bb 185 1 (𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∨ wo 639   = wceq 1259   ∈ wcel 1409   ∪ cun 2942  {csn 3402  suc csuc 4129 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-suc 4135 This theorem is referenced by:  elsuc2  4171  nntri3or  6102  frec2uzltd  9347
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