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Theorem bj-inex 9292
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 2562 . 2 (A 𝑉x x = A)
2 elisset 2562 . 2 (B 𝑊y y = B)
3 ax-17 1416 . . . 4 (y y = Bxy y = B)
4 19.29r 1509 . . . 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 1416 . . . . 5 (x = Ay x = A)
7 19.29 1508 . . . . 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 1488 . . 3 (x(x = A y y = B) → xy(x = A y = B))
10 ineq12 3127 . . . . 5 ((x = A y = B) → (xy) = (AB))
11102eximi 1489 . . . 4 (xy(x = A y = B) → xy(xy) = (AB))
12 dfin5 2919 . . . . . . 7 (xy) = {z xz y}
13 vex 2554 . . . . . . . 8 x V
14 ax-bdel 9210 . . . . . . . . 9 BOUNDED z y
15 bdcv 9237 . . . . . . . . 9 BOUNDED x
1614, 15bdrabexg 9291 . . . . . . . 8 (x V → {z xz y} V)
1713, 16ax-mp 7 . . . . . . 7 {z xz y} V
1812, 17eqeltri 2107 . . . . . 6 (xy) V
19 eleq1 2097 . . . . . 6 ((xy) = (AB) → ((xy) V ↔ (AB) V))
2018, 19mpbii 136 . . . . 5 ((xy) = (AB) → (AB) V)
2120exlimivv 1773 . . . 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 1240   = wceq 1242  wex 1378   wcel 1390  {crab 2304  Vcvv 2551  cin 2910
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9202  ax-bdan 9204  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-bdc 9230
This theorem is referenced by:  speano5  9332
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