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Mirrors > Home > MPE Home > Th. List > disjor | Structured version Visualization version Unicode version |
Description: Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjor.1 |
Ref | Expression |
---|---|
disjor | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disj 4621 | . 2 Disj | |
2 | ralcom4 3224 | . . 3 | |
3 | orcom 402 | . . . . . . 7 | |
4 | df-or 385 | . . . . . . 7 | |
5 | neq0 3930 | . . . . . . . . . 10 | |
6 | elin 3796 | . . . . . . . . . . 11 | |
7 | 6 | exbii 1774 | . . . . . . . . . 10 |
8 | 5, 7 | bitri 264 | . . . . . . . . 9 |
9 | 8 | imbi1i 339 | . . . . . . . 8 |
10 | 19.23v 1902 | . . . . . . . 8 | |
11 | 9, 10 | bitr4i 267 | . . . . . . 7 |
12 | 3, 4, 11 | 3bitri 286 | . . . . . 6 |
13 | 12 | ralbii 2980 | . . . . 5 |
14 | ralcom4 3224 | . . . . 5 | |
15 | 13, 14 | bitri 264 | . . . 4 |
16 | 15 | ralbii 2980 | . . 3 |
17 | disjor.1 | . . . . . 6 | |
18 | 17 | eleq2d 2687 | . . . . 5 |
19 | 18 | rmo4 3399 | . . . 4 |
20 | 19 | albii 1747 | . . 3 |
21 | 2, 16, 20 | 3bitr4i 292 | . 2 |
22 | 1, 21 | bitr4i 267 | 1 Disj |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 wral 2912 wrmo 2915 cin 3573 c0 3915 Disj wdisj 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rmo 2920 df-v 3202 df-dif 3577 df-in 3581 df-nul 3916 df-disj 4621 |
This theorem is referenced by: disjors 4635 disjord 4641 disjiunb 4642 disjxiun 4649 disjxiunOLD 4650 disjxun 4651 otsndisj 4979 otiunsndisj 4980 qsdisj2 7825 s3sndisj 13706 s3iunsndisj 13707 cshwsdisj 15805 dyadmbl 23368 numedglnl 26039 clwwlksndisj 26973 2wspmdisj 27201 disjnf 29384 disjorsf 29393 poimirlem26 33435 mblfinlem2 33447 ndisj2 39218 nnfoctbdjlem 40672 iundjiun 40677 otiunsndisjX 41298 |
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