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Theorem sucprcreg 4300
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 4176 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 4293 . . . 4 ¬ 𝐴𝐴
3 nfv 1437 . . . . 5 𝑥 𝐴𝐴
4 eleq1 2116 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
53, 4ceqsalg 2599 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥𝐴) ↔ 𝐴𝐴))
62, 5mtbiri 610 . . 3 (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴𝑥𝐴))
7 velsn 3419 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 olc 642 . . . . . 6 (𝑥 ∈ {𝐴} → (𝑥𝐴𝑥 ∈ {𝐴}))
9 elun 3111 . . . . . . 7 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
10 ssid 2991 . . . . . . . . 9 𝐴𝐴
11 df-suc 4135 . . . . . . . . . . 11 suc 𝐴 = (𝐴 ∪ {𝐴})
1211eqeq1i 2063 . . . . . . . . . 10 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13 sseq1 2993 . . . . . . . . . 10 ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1412, 13sylbi 118 . . . . . . . . 9 (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1510, 14mpbiri 161 . . . . . . . 8 (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
1615sseld 2971 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥𝐴))
179, 16syl5bir 146 . . . . . 6 (suc 𝐴 = 𝐴 → ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝑥𝐴))
188, 17syl5 32 . . . . 5 (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥𝐴))
197, 18syl5bir 146 . . . 4 (suc 𝐴 = 𝐴 → (𝑥 = 𝐴𝑥𝐴))
2019alrimiv 1770 . . 3 (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥𝐴))
216, 20nsyl3 566 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
221, 21impbii 121 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574  cun 2942  wss 2944  {csn 3402  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-suc 4135
This theorem is referenced by: (None)
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