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Mirrors > Home > MPE Home > Th. List > Mathboxes > idsset | Structured version Visualization version Unicode version |
Description: is equal to the intersection of and its converse. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
idsset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5249 | . 2 | |
2 | relsset 31995 | . . 3 | |
3 | relin1 5236 | . . 3 | |
4 | 2, 3 | ax-mp 5 | . 2 |
5 | eqss 3618 | . . 3 | |
6 | vex 3203 | . . . 4 | |
7 | 6 | ideq 5274 | . . 3 |
8 | brin 4704 | . . . 4 | |
9 | 6 | brsset 31996 | . . . . 5 |
10 | vex 3203 | . . . . . . 7 | |
11 | 10, 6 | brcnv 5305 | . . . . . 6 |
12 | 10 | brsset 31996 | . . . . . 6 |
13 | 11, 12 | bitri 264 | . . . . 5 |
14 | 9, 13 | anbi12i 733 | . . . 4 |
15 | 8, 14 | bitri 264 | . . 3 |
16 | 5, 7, 15 | 3bitr4i 292 | . 2 |
17 | 1, 4, 16 | eqbrriv 5215 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 cin 3573 wss 3574 class class class wbr 4653 cid 5023 ccnv 5113 wrel 5119 csset 31939 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-eprel 5029 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-sset 31963 |
This theorem is referenced by: (None) |
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