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Theorem bdinex1 13024
Description: Bounded version of inex1 4032. (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 12971 . . . . 5 BOUNDED 𝑦𝐵
43bdzfauscl 13015 . . . 4 (𝐴 ∈ V → ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
51, 4ax-mp 5 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
6 dfcleq 2111 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
7 elin 3229 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
87bibi2i 226 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
98albii 1431 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
106, 9bitri 183 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
1110exbii 1569 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
125, 11mpbir 145 . 2 𝑥 𝑥 = (𝐴𝐵)
1312issetri 2669 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1314   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  cin 3040  BOUNDED wbdc 12965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-bdc 12966
This theorem is referenced by:  bdinex2  13025  bdinex1g  13026  bdpeano5  13068
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