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Mirrors > Home > MPE Home > Th. List > ecopovsym | Structured version Visualization version Unicode version |
Description: Assuming the operation is commutative, show that the relation , specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopopr.com |
Ref | Expression |
---|---|
ecopovsym |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . 5 | |
2 | opabssxp 5193 | . . . . 5 | |
3 | 1, 2 | eqsstri 3635 | . . . 4 |
4 | 3 | brel 5168 | . . 3 |
5 | eqid 2622 | . . . 4 | |
6 | breq1 4656 | . . . . 5 | |
7 | breq2 4657 | . . . . 5 | |
8 | 6, 7 | bibi12d 335 | . . . 4 |
9 | breq2 4657 | . . . . 5 | |
10 | breq1 4656 | . . . . 5 | |
11 | 9, 10 | bibi12d 335 | . . . 4 |
12 | 1 | ecopoveq 7848 | . . . . . 6 |
13 | vex 3203 | . . . . . . . . 9 | |
14 | vex 3203 | . . . . . . . . 9 | |
15 | ecopopr.com | . . . . . . . . 9 | |
16 | 13, 14, 15 | caovcom 6831 | . . . . . . . 8 |
17 | vex 3203 | . . . . . . . . 9 | |
18 | vex 3203 | . . . . . . . . 9 | |
19 | 17, 18, 15 | caovcom 6831 | . . . . . . . 8 |
20 | 16, 19 | eqeq12i 2636 | . . . . . . 7 |
21 | eqcom 2629 | . . . . . . 7 | |
22 | 20, 21 | bitri 264 | . . . . . 6 |
23 | 12, 22 | syl6bb 276 | . . . . 5 |
24 | 1 | ecopoveq 7848 | . . . . . 6 |
25 | 24 | ancoms 469 | . . . . 5 |
26 | 23, 25 | bitr4d 271 | . . . 4 |
27 | 5, 8, 11, 26 | 2optocl 5196 | . . 3 |
28 | 4, 27 | syl 17 | . 2 |
29 | 28 | ibi 256 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cop 4183 class class class wbr 4653 copab 4712 cxp 5112 (class class class)co 6650 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 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-xp 5120 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: ecopover 7851 ecopoverOLD 7852 |
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