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Theorem bj-sucexg 7145
Description: sucexg 4174 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sucexg (A 𝑉 → suc A V)

Proof of Theorem bj-sucexg
StepHypRef Expression
1 bj-snexg 7135 . . . 4 (A 𝑉 → {A} V)
21pm4.71i 371 . . 3 (A 𝑉 ↔ (A 𝑉 {A} V))
32biimpi 113 . 2 (A 𝑉 → (A 𝑉 {A} V))
4 bj-unexg 7144 . 2 ((A 𝑉 {A} V) → (A ∪ {A}) V)
5 df-suc 4057 . . . 4 suc A = (A ∪ {A})
65eleq1i 2085 . . 3 (suc A V ↔ (A ∪ {A}) V)
76biimpri 124 . 2 ((A ∪ {A}) V → suc A V)
83, 4, 73syl 17 1 (A 𝑉 → suc A V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  Vcvv 2535  cun 2892  {csn 3350  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-suc 4057  df-bdc 7068
This theorem is referenced by:  bj-sucex  7146
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