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Theorem bj-inex 10386
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 2585 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2585 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 ax-17 1435 . . . 4 (∃𝑦 𝑦 = 𝐵 → ∀𝑥𝑦 𝑦 = 𝐵)
4 19.29r 1528 . . . 4 ((∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylan2 274 . . 3 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 ax-17 1435 . . . . 5 (𝑥 = 𝐴 → ∀𝑦 𝑥 = 𝐴)
7 19.29 1527 . . . . 5 ((∀𝑦 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
86, 7sylan 271 . . . 4 ((𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑦(𝑥 = 𝐴𝑦 = 𝐵))
98eximi 1507 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
10 ineq12 3160 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
11102eximi 1508 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝑥𝑦) = (𝐴𝐵))
12 dfin5 2952 . . . . . . 7 (𝑥𝑦) = {𝑧𝑥𝑧𝑦}
13 vex 2577 . . . . . . . 8 𝑥 ∈ V
14 ax-bdel 10300 . . . . . . . . 9 BOUNDED 𝑧𝑦
15 bdcv 10327 . . . . . . . . 9 BOUNDED 𝑥
1614, 15bdrabexg 10385 . . . . . . . 8 (𝑥 ∈ V → {𝑧𝑥𝑧𝑦} ∈ V)
1713, 16ax-mp 7 . . . . . . 7 {𝑧𝑥𝑧𝑦} ∈ V
1812, 17eqeltri 2126 . . . . . 6 (𝑥𝑦) ∈ V
19 eleq1 2116 . . . . . 6 ((𝑥𝑦) = (𝐴𝐵) → ((𝑥𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
2018, 19mpbii 140 . . . . 5 ((𝑥𝑦) = (𝐴𝐵) → (𝐴𝐵) ∈ V)
2120exlimivv 1792 . . . 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 277 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  wcel 1409  {crab 2327  Vcvv 2574  cin 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bd0 10292  ax-bdan 10294  ax-bdel 10300  ax-bdsb 10301  ax-bdsep 10363
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-in 2951  df-ss 2958  df-bdc 10320
This theorem is referenced by:  speano5  10428
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