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Theorem bj-inex 12939
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 2672 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2672 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 ax-17 1489 . . . 4 (∃𝑦 𝑦 = 𝐵 → ∀𝑥𝑦 𝑦 = 𝐵)
4 19.29r 1583 . . . 4 ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylan2 282 . . 3 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 ax-17 1489 . . . . 5 (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7 19.29 1582 . . . . 5 ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
86, 7sylan 279 . . . 4 ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
98eximi 1562 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 ineq12 3240 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
11102eximi 1563 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝑥𝑦) = (𝐴𝐵))
12 dfin5 3046 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
13 vex 2661 . . . . . . . 8 𝑥 ∈ V
14 ax-bdel 12853 . . . . . . . . 9 BOUNDED 𝑧𝑦
15 bdcv 12880 . . . . . . . . 9 BOUNDED 𝑥
1614, 15bdrabexg 12938 . . . . . . . 8 (𝑥 ∈ V → {𝑧𝑥𝑧𝑦} ∈ V)
1713, 16ax-mp 5 . . . . . . 7 {𝑧𝑥𝑧𝑦} ∈ V
1812, 17eqeltri 2188 . . . . . 6 (𝑥𝑦) ∈ V
19 eleq1 2178 . . . . . 6 ((𝑥𝑦) = (𝐴𝐵) → ((𝑥𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
2018, 19mpbii 147 . . . . 5 ((𝑥𝑦) = (𝐴𝐵) → (𝐴𝐵) ∈ V)
2120exlimivv 1850 . . . 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 285 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wex 1451  wcel 1463  {crab 2395  Vcvv 2658  cin 3038
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12845  ax-bdan 12847  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-bdc 12873
This theorem is referenced by:  speano5  12976
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