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Mirrors > Home > ILE Home > Th. List > intirr | Unicode version |
Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
intirr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3319 | . . . 4 | |
2 | 1 | eqeq1i 2178 | . . 3 |
3 | disj2 3469 | . . 3 | |
4 | reli 4738 | . . . 4 | |
5 | ssrel 4697 | . . . 4 | |
6 | 4, 5 | ax-mp 5 | . . 3 |
7 | 2, 3, 6 | 3bitri 205 | . 2 |
8 | equcom 1699 | . . . . 5 | |
9 | vex 2733 | . . . . . 6 | |
10 | 9 | ideq 4761 | . . . . 5 |
11 | df-br 3988 | . . . . 5 | |
12 | 8, 10, 11 | 3bitr2i 207 | . . . 4 |
13 | vex 2733 | . . . . . . . 8 | |
14 | 13, 9 | opex 4212 | . . . . . . 7 |
15 | 14 | biantrur 301 | . . . . . 6 |
16 | eldif 3130 | . . . . . 6 | |
17 | 15, 16 | bitr4i 186 | . . . . 5 |
18 | df-br 3988 | . . . . 5 | |
19 | 17, 18 | xchnxbir 676 | . . . 4 |
20 | 12, 19 | imbi12i 238 | . . 3 |
21 | 20 | 2albii 1464 | . 2 |
22 | nfv 1521 | . . . 4 | |
23 | breq2 3991 | . . . . 5 | |
24 | 23 | notbid 662 | . . . 4 |
25 | 22, 24 | equsal 1720 | . . 3 |
26 | 25 | albii 1463 | . 2 |
27 | 7, 21, 26 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1346 wceq 1348 wcel 2141 cvv 2730 cdif 3118 cin 3120 wss 3121 c0 3414 cop 3584 class class class wbr 3987 cid 4271 wrel 4614 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 |
This theorem is referenced by: (None) |
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