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Theorem sucprcreg 4544
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 4408 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 4536 . . . 4 ¬ 𝐴𝐴
3 nfv 1528 . . . . 5 𝑥 𝐴𝐴
4 eleq1 2240 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
53, 4ceqsalg 2765 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥𝐴) ↔ 𝐴𝐴))
62, 5mtbiri 675 . . 3 (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴𝑥𝐴))
7 velsn 3608 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 olc 711 . . . . . 6 (𝑥 ∈ {𝐴} → (𝑥𝐴𝑥 ∈ {𝐴}))
9 elun 3276 . . . . . . 7 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
10 ssid 3175 . . . . . . . . 9 𝐴𝐴
11 df-suc 4367 . . . . . . . . . . 11 suc 𝐴 = (𝐴 ∪ {𝐴})
1211eqeq1i 2185 . . . . . . . . . 10 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13 sseq1 3178 . . . . . . . . . 10 ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1412, 13sylbi 121 . . . . . . . . 9 (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1510, 14mpbiri 168 . . . . . . . 8 (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
1615sseld 3154 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥𝐴))
179, 16biimtrrid 153 . . . . . 6 (suc 𝐴 = 𝐴 → ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝑥𝐴))
188, 17syl5 32 . . . . 5 (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥𝐴))
197, 18biimtrrid 153 . . . 4 (suc 𝐴 = 𝐴 → (𝑥 = 𝐴𝑥𝐴))
2019alrimiv 1874 . . 3 (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥𝐴))
216, 20nsyl3 626 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
221, 21impbii 126 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  wss 3129  {csn 3591  suc csuc 4361
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-suc 4367
This theorem is referenced by: (None)
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