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Theorem disjorf 23974
 Description: Two ways to say that a collection for is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
disjorf.1
disjorf.2
disjorf.3
Assertion
Ref Expression
disjorf Disj
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem disjorf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-disj 4143 . 2 Disj
2 ralcom4 2934 . . 3
3 orcom 377 . . . . . . 7
4 df-or 360 . . . . . . 7
5 neq0 3598 . . . . . . . . . 10
6 elin 3490 . . . . . . . . . . 11
76exbii 1589 . . . . . . . . . 10
85, 7bitri 241 . . . . . . . . 9
98imbi1i 316 . . . . . . . 8
10 19.23v 1910 . . . . . . . 8
119, 10bitr4i 244 . . . . . . 7
123, 4, 113bitri 263 . . . . . 6
1312ralbii 2690 . . . . 5
14 ralcom4 2934 . . . . 5
1513, 14bitri 241 . . . 4
1615ralbii 2690 . . 3
17 disjorf.1 . . . . 5
18 disjorf.2 . . . . 5
19 nfv 1626 . . . . 5
20 disjorf.3 . . . . . 6
2120eleq2d 2471 . . . . 5
2217, 18, 19, 21rmo4f 23937 . . . 4
2322albii 1572 . . 3
242, 16, 233bitr4i 269 . 2
251, 24bitr4i 244 1 Disj
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1546  wex 1547   wceq 1649   wcel 1721  wnfc 2527  wral 2666  wrmo 2669   cin 3279  c0 3588  Disj wdisj 4142 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rmo 2674  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589  df-disj 4143
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