Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdinex1 GIF version

Theorem bdinex1 9892
Description: Bounded version of inex1 3888. (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 9839 . . . . 5 BOUNDED 𝑦𝐵
43bdzfauscl 9883 . . . 4 (𝐴 ∈ V → ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
51, 4ax-mp 7 . . 3 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵))
6 dfcleq 2034 . . . . 5 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)))
7 elin 3123 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
87bibi2i 216 . . . . . 6 ((𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ (𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
98albii 1359 . . . . 5 (∀𝑦(𝑦𝑥𝑦 ∈ (𝐴𝐵)) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
106, 9bitri 173 . . . 4 (𝑥 = (𝐴𝐵) ↔ ∀𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
1110exbii 1496 . . 3 (∃𝑥 𝑥 = (𝐴𝐵) ↔ ∃𝑥𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦𝐵)))
125, 11mpbir 134 . 2 𝑥 𝑥 = (𝐴𝐵)
1312issetri 2561 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wal 1241   = wceq 1243  wex 1381  wcel 1393  Vcvv 2554  cin 2913  BOUNDED wbdc 9833
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 9877
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-bdc 9834
This theorem is referenced by:  bdinex2  9893  bdinex1g  9894  bdpeano5  9941
  Copyright terms: Public domain W3C validator