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Theorem sucprcreg 4424
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 4294 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 4416 . . . 4 ¬ 𝐴𝐴
3 nfv 1491 . . . . 5 𝑥 𝐴𝐴
4 eleq1 2177 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
53, 4ceqsalg 2685 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥𝐴) ↔ 𝐴𝐴))
62, 5mtbiri 647 . . 3 (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴𝑥𝐴))
7 velsn 3510 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 olc 683 . . . . . 6 (𝑥 ∈ {𝐴} → (𝑥𝐴𝑥 ∈ {𝐴}))
9 elun 3183 . . . . . . 7 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
10 ssid 3083 . . . . . . . . 9 𝐴𝐴
11 df-suc 4253 . . . . . . . . . . 11 suc 𝐴 = (𝐴 ∪ {𝐴})
1211eqeq1i 2122 . . . . . . . . . 10 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13 sseq1 3086 . . . . . . . . . 10 ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1412, 13sylbi 120 . . . . . . . . 9 (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1510, 14mpbiri 167 . . . . . . . 8 (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
1615sseld 3062 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥𝐴))
179, 16syl5bir 152 . . . . . 6 (suc 𝐴 = 𝐴 → ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝑥𝐴))
188, 17syl5 32 . . . . 5 (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥𝐴))
197, 18syl5bir 152 . . . 4 (suc 𝐴 = 𝐴 → (𝑥 = 𝐴𝑥𝐴))
2019alrimiv 1828 . . 3 (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥𝐴))
216, 20nsyl3 598 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
221, 21impbii 125 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680  wal 1312   = wceq 1314  wcel 1463  Vcvv 2657  cun 3035  wss 3037  {csn 3493  suc csuc 4247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-sn 3499  df-suc 4253
This theorem is referenced by: (None)
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