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Theorem disjxp1 6177
 Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
disjxp1.1 Disj
Assertion
Ref Expression
disjxp1 Disj
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem disjxp1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6107 . . . . . . 7
2 xp1st 6107 . . . . . . 7
3 disjxp1.1 . . . . . . . . . . . 12 Disj
4 df-disj 3943 . . . . . . . . . . . 12 Disj
53, 4sylib 121 . . . . . . . . . . 11
6 1stexg 6109 . . . . . . . . . . . . 13
76elv 2716 . . . . . . . . . . . 12
8 eleq1 2220 . . . . . . . . . . . . 13
98rmobidv 2645 . . . . . . . . . . . 12
107, 9spcv 2806 . . . . . . . . . . 11
115, 10syl 14 . . . . . . . . . 10
12 nfcv 2299 . . . . . . . . . . 11
13 nfcv 2299 . . . . . . . . . . 11
14 nfcsb1v 3064 . . . . . . . . . . . 12
1514nfel2 2312 . . . . . . . . . . 11
16 csbeq1a 3040 . . . . . . . . . . . 12
1716eleq2d 2227 . . . . . . . . . . 11
1812, 13, 15, 17rmo4f 2910 . . . . . . . . . 10
1911, 18sylib 121 . . . . . . . . 9
2019r19.21bi 2545 . . . . . . . 8
2120r19.21bi 2545 . . . . . . 7
221, 2, 21syl2ani 406 . . . . . 6
2322ralrimiva 2530 . . . . 5
2423ralrimiva 2530 . . . 4
25 nfcsb1v 3064 . . . . . . 7
2614, 25nfxp 4610 . . . . . 6
2726nfel2 2312 . . . . 5
28 csbeq1a 3040 . . . . . . 7
2916, 28xpeq12d 4608 . . . . . 6
3029eleq2d 2227 . . . . 5
3112, 13, 27, 30rmo4f 2910 . . . 4
3224, 31sylibr 133 . . 3
3332alrimiv 1854 . 2
34 df-disj 3943 . 2 Disj
3533, 34sylibr 133 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103  wal 1333   wceq 1335   wcel 2128  wral 2435  wrmo 2438  cvv 2712  csb 3031  Disj wdisj 3942   cxp 4581  cfv 5167  c1st 6080 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rmo 2443  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fo 5173  df-fv 5175  df-1st 6082 This theorem is referenced by:  disjsnxp  6178
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