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Theorem bdinex1 11447
Description: Bounded version of inex1 3965. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdinex1.bd BOUNDED 𝐵
bdinex1.1 𝐴 ∈ V
Assertion
Ref Expression
bdinex1 (𝐴𝐵) ∈ V

Proof of Theorem bdinex1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdinex1.1 . . . 4 𝐴 ∈ V
2 bdinex1.bd . . . . . 6 BOUNDED 𝐵
32bdeli 11394 . . . . 5 BOUNDED 𝑦𝐵
43bdzfauscl 11438 . . . 4 (𝐴 ∈ V → ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
51, 4ax-mp 7 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
6 dfcleq 2082 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
7 elin 3181 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
87bibi2i 225 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
98albii 1404 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
106, 9bitri 182 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
1110exbii 1541 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
125, 11mpbir 144 . 2 𝑥 𝑥 = (𝐴𝐵)
1312issetri 2628 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cin 2996  BOUNDED wbdc 11388
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-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-bdsep 11432
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-bdc 11389
This theorem is referenced by:  bdinex2  11448  bdinex1g  11449  bdpeano5  11495
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