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Mirrors > Home > ILE Home > Th. List > erth | Unicode version |
Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
erth.1 | |
erth.2 |
Ref | Expression |
---|---|
erth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . 7 | |
2 | erth.1 | . . . . . . . . 9 | |
3 | 2 | ersymb 6527 | . . . . . . . 8 |
4 | 3 | biimpa 294 | . . . . . . 7 |
5 | 1, 4 | jca 304 | . . . . . 6 |
6 | 2 | ertr 6528 | . . . . . . 7 |
7 | 6 | impl 378 | . . . . . 6 |
8 | 5, 7 | sylan 281 | . . . . 5 |
9 | 2 | ertr 6528 | . . . . . 6 |
10 | 9 | impl 378 | . . . . 5 |
11 | 8, 10 | impbida 591 | . . . 4 |
12 | vex 2733 | . . . . 5 | |
13 | erth.2 | . . . . . 6 | |
14 | 13 | adantr 274 | . . . . 5 |
15 | elecg 6551 | . . . . 5 | |
16 | 12, 14, 15 | sylancr 412 | . . . 4 |
17 | errel 6522 | . . . . . . 7 | |
18 | 2, 17 | syl 14 | . . . . . 6 |
19 | brrelex2 4652 | . . . . . 6 | |
20 | 18, 19 | sylan 281 | . . . . 5 |
21 | elecg 6551 | . . . . 5 | |
22 | 12, 20, 21 | sylancr 412 | . . . 4 |
23 | 11, 16, 22 | 3bitr4d 219 | . . 3 |
24 | 23 | eqrdv 2168 | . 2 |
25 | 2 | adantr 274 | . . 3 |
26 | 2, 13 | erref 6533 | . . . . . . 7 |
27 | 26 | adantr 274 | . . . . . 6 |
28 | 13 | adantr 274 | . . . . . . 7 |
29 | elecg 6551 | . . . . . . 7 | |
30 | 28, 28, 29 | syl2anc 409 | . . . . . 6 |
31 | 27, 30 | mpbird 166 | . . . . 5 |
32 | simpr 109 | . . . . 5 | |
33 | 31, 32 | eleqtrd 2249 | . . . 4 |
34 | 25, 32 | ereldm 6556 | . . . . . 6 |
35 | 28, 34 | mpbid 146 | . . . . 5 |
36 | elecg 6551 | . . . . 5 | |
37 | 28, 35, 36 | syl2anc 409 | . . . 4 |
38 | 33, 37 | mpbid 146 | . . 3 |
39 | 25, 38 | ersym 6525 | . 2 |
40 | 24, 39 | impbida 591 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 class class class wbr 3989 wrel 4616 wer 6510 cec 6511 |
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-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 4107 ax-pow 4160 ax-pr 4194 |
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-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-er 6513 df-ec 6515 |
This theorem is referenced by: erth2 6558 erthi 6559 qliftfun 6595 eroveu 6604 th3qlem1 6615 enqeceq 7321 enq0eceq 7399 nnnq0lem1 7408 enreceq 7698 prsrlem1 7704 |
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