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Theorem disjprg 4448
Description: A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjprg.1
disjprg.2
Assertion
Ref Expression
disjprg
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem disjprg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . . 7
2 nfcv 2619 . . . . . . . . . 10
3 nfcv 2619 . . . . . . . . . 10
4 disjprg.1 . . . . . . . . . 10
52, 3, 4csbhypf 3453 . . . . . . . . 9
65ineq1d 3698 . . . . . . . 8
76eqeq1d 2459 . . . . . . 7
81, 7orbi12d 709 . . . . . 6
98ralbidv 2896 . . . . 5
10 eqeq1 2461 . . . . . . 7
11 nfcv 2619 . . . . . . . . . 10
12 nfcv 2619 . . . . . . . . . 10
13 disjprg.2 . . . . . . . . . 10
1411, 12, 13csbhypf 3453 . . . . . . . . 9
1514ineq1d 3698 . . . . . . . 8
1615eqeq1d 2459 . . . . . . 7
1710, 16orbi12d 709 . . . . . 6
1817ralbidv 2896 . . . . 5
199, 18ralprg 4078 . . . 4
20193adant3 1016 . . 3
21 id 22 . . . . . . . . . 10
2221eqcomd 2465 . . . . . . . . 9
2322orcd 392 . . . . . . . 8
24 a1tru 1411 . . . . . . . 8
2523, 242thd 240 . . . . . . 7
26 eqeq2 2472 . . . . . . . 8
2711, 12, 13csbhypf 3453 . . . . . . . . . 10
2827ineq2d 3699 . . . . . . . . 9
2928eqeq1d 2459 . . . . . . . 8
3026, 29orbi12d 709 . . . . . . 7
3125, 30ralprg 4078 . . . . . 6
32313adant3 1016 . . . . 5
33 simp3 998 . . . . . . . 8
3433neneqd 2659 . . . . . . 7
35 biorf 405 . . . . . . 7
3634, 35syl 16 . . . . . 6
37 tru 1399 . . . . . . 7
3837biantrur 506 . . . . . 6
3936, 38syl6bb 261 . . . . 5
4032, 39bitr4d 256 . . . 4
41 eqeq2 2472 . . . . . . . . 9
42 eqcom 2466 . . . . . . . . 9
4341, 42syl6bb 261 . . . . . . . 8
442, 3, 4csbhypf 3453 . . . . . . . . . . 11
4544ineq2d 3699 . . . . . . . . . 10
46 incom 3690 . . . . . . . . . 10
4745, 46syl6eq 2514 . . . . . . . . 9
4847eqeq1d 2459 . . . . . . . 8
4943, 48orbi12d 709 . . . . . . 7
50 id 22 . . . . . . . . . 10
5150eqcomd 2465 . . . . . . . . 9
5251orcd 392 . . . . . . . 8
53 a1tru 1411 . . . . . . . 8
5452, 532thd 240 . . . . . . 7
5549, 54ralprg 4078 . . . . . 6
56553adant3 1016 . . . . 5
5737biantru 505 . . . . . 6
5836, 57syl6bb 261 . . . . 5
5956, 58bitr4d 256 . . . 4
6040, 59anbi12d 710 . . 3
6120, 60bitrd 253 . 2
62 disjors 4438 . 2
63 pm4.24 643 . 2
6461, 62, 633bitr4g 288 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  /\w3a 973  =wceq 1395   wtru 1396  e.wcel 1818  =/=wne 2652  A.wral 2807  [_csb 3434  i^icin 3474   c0 3784  {cpr 4031  Disj_wdisj 4422
This theorem is referenced by:  disjdifprg  27436  probun  28358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-nul 3785  df-sn 4030  df-pr 4032  df-disj 4423
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