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Mirrors > Home > NFE Home > Th. List > disjex | Unicode version |
Description: The disjointedness relationship is a set. (Contributed by SF, 11-Feb-2015.) |
Ref | Expression |
---|---|
disjex | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disj 5744 | . . 3 Disj | |
2 | oteltxp 5782 | . . . . . . . . . 10 S S S S | |
3 | vex 2862 | . . . . . . . . . . . 12 | |
4 | vex 2862 | . . . . . . . . . . . 12 | |
5 | 3, 4 | opelssetsn 4760 | . . . . . . . . . . 11 S |
6 | vex 2862 | . . . . . . . . . . . 12 | |
7 | 3, 6 | opelssetsn 4760 | . . . . . . . . . . 11 S |
8 | 5, 7 | anbi12i 678 | . . . . . . . . . 10 S S |
9 | 2, 8 | bitri 240 | . . . . . . . . 9 S S |
10 | 9 | exbii 1582 | . . . . . . . 8 S S |
11 | elima1c 4947 | . . . . . . . 8 S S 1c S S | |
12 | df-rex 2620 | . . . . . . . 8 | |
13 | 10, 11, 12 | 3bitr4i 268 | . . . . . . 7 S S 1c |
14 | dfrex2 2627 | . . . . . . 7 | |
15 | 13, 14 | bitri 240 | . . . . . 6 S S 1c |
16 | 15 | con2bii 322 | . . . . 5 S S 1c |
17 | disj 3591 | . . . . 5 | |
18 | 4, 6 | opex 4588 | . . . . . 6 |
19 | 18 | elcompl 3225 | . . . . 5 ∼ S S 1c S S 1c |
20 | 16, 17, 19 | 3bitr4ri 269 | . . . 4 ∼ S S 1c |
21 | 20 | opabbi2i 4866 | . . 3 ∼ S S 1c |
22 | 1, 21 | eqtr4i 2376 | . 2 Disj ∼ S S 1c |
23 | ssetex 4744 | . . . . 5 S | |
24 | 23, 23 | txpex 5785 | . . . 4 S S |
25 | 1cex 4142 | . . . 4 1c | |
26 | 24, 25 | imaex 4747 | . . 3 S S 1c |
27 | 26 | complex 4104 | . 2 ∼ S S 1c |
28 | 22, 27 | eqeltri 2423 | 1 Disj |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wa 358 wex 1541 wceq 1642 wcel 1710 wral 2614 wrex 2615 cvv 2859 ∼ ccompl 3205 cin 3208 c0 3550 csn 3737 1cc1c 4134 cop 4561 copab 4622 S csset 4719 cima 4722 ctxp 5735 Disj cdisj 5743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-cnv 4785 df-2nd 4797 df-txp 5736 df-disj 5744 |
This theorem is referenced by: addcfnex 5824 |
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