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Theorem bdinex1 13934
Description: Bounded version of inex1 4123. (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 13881 . . . . 5 BOUNDED 𝑦𝐵
43bdzfauscl 13925 . . . 4 (𝐴 ∈ V → ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
51, 4ax-mp 5 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
6 dfcleq 2164 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
7 elin 3310 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
87bibi2i 226 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
98albii 1463 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
106, 9bitri 183 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
1110exbii 1598 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
125, 11mpbir 145 . 2 𝑥 𝑥 = (𝐴𝐵)
1312issetri 2739 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1346   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  cin 3120  BOUNDED wbdc 13875
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-bdc 13876
This theorem is referenced by:  bdinex2  13935  bdinex1g  13936  bdpeano5  13978
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