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Theorem elsuci 4450
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 → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4418 . . . 4 suc 𝐵 = (𝐵 ∪ {𝐵})
21eleq2i 2272 . . 3 (𝐴 ∈ suc 𝐵𝐴 ∈ (𝐵 ∪ {𝐵}))
3 elun 3314 . . 3 (𝐴 ∈ (𝐵 ∪ {𝐵}) ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
42, 3bitri 184 . 2 (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 ∈ {𝐵}))
5 elsni 3651 . . 3 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
65orim2i 763 . 2 ((𝐴𝐵𝐴 ∈ {𝐵}) → (𝐴𝐵𝐴 = 𝐵))
74, 6sylbi 121 1 (𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 710   = wceq 1373  wcel 2176  cun 3164  {csn 3633  suc csuc 4412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-suc 4418
This theorem is referenced by:  trsucss  4470  onsucelsucexmid  4578  ordsoexmid  4610  ordsuc  4611  ordpwsucexmid  4618  nnsucelsuc  6577  nntri3or  6579  nnmordi  6602  nnaordex  6614  phplem3  6951  nninfninc  7225  nnnninf2  7229  3nelsucpw1  7346  3nsssucpw1  7348
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