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Theorem sucprcreg 4355
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
Assertion
Ref Expression
sucprcreg 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Proof of Theorem sucprcreg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sucprc 4230 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 4347 . . . 4 ¬ 𝐴𝐴
3 nfv 1466 . . . . 5 𝑥 𝐴𝐴
4 eleq1 2150 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
53, 4ceqsalg 2647 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥𝐴) ↔ 𝐴𝐴))
62, 5mtbiri 635 . . 3 (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴𝑥𝐴))
7 velsn 3458 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 olc 667 . . . . . 6 (𝑥 ∈ {𝐴} → (𝑥𝐴𝑥 ∈ {𝐴}))
9 elun 3139 . . . . . . 7 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
10 ssid 3042 . . . . . . . . 9 𝐴𝐴
11 df-suc 4189 . . . . . . . . . . 11 suc 𝐴 = (𝐴 ∪ {𝐴})
1211eqeq1i 2095 . . . . . . . . . 10 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13 sseq1 3045 . . . . . . . . . 10 ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1412, 13sylbi 119 . . . . . . . . 9 (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1510, 14mpbiri 166 . . . . . . . 8 (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
1615sseld 3022 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥𝐴))
179, 16syl5bir 151 . . . . . 6 (suc 𝐴 = 𝐴 → ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝑥𝐴))
188, 17syl5 32 . . . . 5 (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥𝐴))
197, 18syl5bir 151 . . . 4 (suc 𝐴 = 𝐴 → (𝑥 = 𝐴𝑥𝐴))
2019alrimiv 1802 . . 3 (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥𝐴))
216, 20nsyl3 591 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
221, 21impbii 124 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  wal 1287   = wceq 1289  wcel 1438  Vcvv 2619  cun 2995  wss 2997  {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-in1 579  ax-in2 580  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  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-suc 4189
This theorem is referenced by: (None)
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