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Theorem bj-inex 14315
Description: The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inex ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem bj-inex
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 2751 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2751 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 ax-17 1526 . . . 4 (∃𝑦 𝑦 = 𝐵 → ∀𝑥𝑦 𝑦 = 𝐵)
4 19.29r 1621 . . . 4 ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylan2 286 . . 3 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 ax-17 1526 . . . . 5 (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7 19.29 1620 . . . . 5 ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
86, 7sylan 283 . . . 4 ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
98eximi 1600 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 ineq12 3331 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
11102eximi 1601 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝑥𝑦) = (𝐴𝐵))
12 dfin5 3136 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
13 vex 2740 . . . . . . . 8 𝑥 ∈ V
14 ax-bdel 14229 . . . . . . . . 9 BOUNDED 𝑧𝑦
15 bdcv 14256 . . . . . . . . 9 BOUNDED 𝑥
1614, 15bdrabexg 14314 . . . . . . . 8 (𝑥 ∈ V → {𝑧𝑥𝑧𝑦} ∈ V)
1713, 16ax-mp 5 . . . . . . 7 {𝑧𝑥𝑧𝑦} ∈ V
1812, 17eqeltri 2250 . . . . . 6 (𝑥𝑦) ∈ V
19 eleq1 2240 . . . . . 6 ((𝑥𝑦) = (𝐴𝐵) → ((𝑥𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
2018, 19mpbii 148 . . . . 5 ((𝑥𝑦) = (𝐴𝐵) → (𝐴𝐵) ∈ V)
2120exlimivv 1896 . . . 4 (∃𝑥𝑦(𝑥𝑦) = (𝐴𝐵) → (𝐴𝐵) ∈ V)
2211, 21syl 14 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
235, 9, 223syl 17 . 2 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
241, 2, 23syl2an 289 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wex 1492  wcel 2148  {crab 2459  Vcvv 2737  cin 3128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14221  ax-bdan 14223  ax-bdel 14229  ax-bdsb 14230  ax-bdsep 14292
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-bdc 14249
This theorem is referenced by:  speano5  14352
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