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Theorem elsuci 4106
Description: Membership in a successor. This one-way implication does not require that either A or B be sets. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci (A suc B → (A B A = B))

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4074 . . . 4 suc B = (B ∪ {B})
21eleq2i 2101 . . 3 (A suc BA (B ∪ {B}))
3 elun 3078 . . 3 (A (B ∪ {B}) ↔ (A B A {B}))
42, 3bitri 173 . 2 (A suc B ↔ (A B A {B}))
5 elsni 3391 . . 3 (A {B} → A = B)
65orim2i 677 . 2 ((A B A {B}) → (A B A = B))
74, 6sylbi 114 1 (A suc B → (A B A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  cun 2909  {csn 3367  suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-suc 4074
This theorem is referenced by:  trsucss  4126  onsucelsucexmid  4215  ordsoexmid  4240  ordsuc  4241  ordpwsucexmid  4246  nnsucelsuc  6009  nntri3or  6011  nnmordi  6025  nnaordex  6036
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