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Theorem bj-inex 7130
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 ((A 𝑉 B 𝑊) → (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 2545 . 2 (A 𝑉x x = A)
2 elisset 2545 . 2 (B 𝑊y y = B)
3 ax-17 1400 . . . 4 (y y = Bxy y = B)
4 19.29r 1494 . . . 4 ((x x = A xy y = B) → x(x = A y y = B))
53, 4sylan2 270 . . 3 ((x x = A y y = B) → x(x = A y y = B))
6 ax-17 1400 . . . . 5 (x = Ay x = A)
7 19.29 1493 . . . . 5 ((y x = A y y = B) → y(x = A y = B))
86, 7sylan 267 . . . 4 ((x = A y y = B) → y(x = A y = B))
98eximi 1473 . . 3 (x(x = A y y = B) → xy(x = A y = B))
10 ineq12 3110 . . . . 5 ((x = A y = B) → (xy) = (AB))
11102eximi 1474 . . . 4 (xy(x = A y = B) → xy(xy) = (AB))
12 dfin5 2902 . . . . . . 7 (xy) = {z xz y}
13 vex 2538 . . . . . . . 8 x V
14 ax-bdel 7048 . . . . . . . . 9 BOUNDED z y
15 bdcv 7075 . . . . . . . . 9 BOUNDED x
1614, 15bdrabexg 7129 . . . . . . . 8 (x V → {z xz y} V)
1713, 16ax-mp 7 . . . . . . 7 {z xz y} V
1812, 17eqeltri 2092 . . . . . 6 (xy) V
19 eleq1 2082 . . . . . 6 ((xy) = (AB) → ((xy) V ↔ (AB) V))
2018, 19mpbii 136 . . . . 5 ((xy) = (AB) → (AB) V)
2120exlimivv 1758 . . . 4 (xy(xy) = (AB) → (AB) V)
2211, 21syl 14 . . 3 (xy(x = A y = B) → (AB) V)
235, 9, 223syl 17 . 2 ((x x = A y y = B) → (AB) V)
241, 2, 23syl2an 273 1 ((A 𝑉 B 𝑊) → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228  wex 1362   wcel 1374  {crab 2288  Vcvv 2535  cin 2893
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7040  ax-bdan 7042  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-bdc 7068
This theorem is referenced by:  speano5  7166
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