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Theorem bj-inex 11798
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 2633 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2633 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 ax-17 1464 . . . 4 (∃𝑦 𝑦 = 𝐵 → ∀𝑥𝑦 𝑦 = 𝐵)
4 19.29r 1557 . . . 4 ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylan2 280 . . 3 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 ax-17 1464 . . . . 5 (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7 19.29 1556 . . . . 5 ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
86, 7sylan 277 . . . 4 ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
98eximi 1536 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 ineq12 3196 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
11102eximi 1537 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝑥𝑦) = (𝐴𝐵))
12 dfin5 3006 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
13 vex 2622 . . . . . . . 8 𝑥 ∈ V
14 ax-bdel 11712 . . . . . . . . 9 BOUNDED 𝑧𝑦
15 bdcv 11739 . . . . . . . . 9 BOUNDED 𝑥
1614, 15bdrabexg 11797 . . . . . . . 8 (𝑥 ∈ V → {𝑧𝑥𝑧𝑦} ∈ V)
1713, 16ax-mp 7 . . . . . . 7 {𝑧𝑥𝑧𝑦} ∈ V
1812, 17eqeltri 2160 . . . . . 6 (𝑥𝑦) ∈ V
19 eleq1 2150 . . . . . 6 ((𝑥𝑦) = (𝐴𝐵) → ((𝑥𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
2018, 19mpbii 146 . . . . 5 ((𝑥𝑦) = (𝐴𝐵) → (𝐴𝐵) ∈ V)
2120exlimivv 1824 . . . 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 283 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  wcel 1438  {crab 2363  Vcvv 2619  cin 2998
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-bd0 11704  ax-bdan 11706  ax-bdel 11712  ax-bdsb 11713  ax-bdsep 11775
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-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-bdc 11732
This theorem is referenced by:  speano5  11839
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