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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 6497 | . . . . . . . 8 |
4 | 3 | biimpa 294 | . . . . . . 7 |
5 | 1, 4 | jca 304 | . . . . . 6 |
6 | 2 | ertr 6498 | . . . . . . 7 |
7 | 6 | impl 378 | . . . . . 6 |
8 | 5, 7 | sylan 281 | . . . . 5 |
9 | 2 | ertr 6498 | . . . . . 6 |
10 | 9 | impl 378 | . . . . 5 |
11 | 8, 10 | impbida 586 | . . . 4 |
12 | vex 2715 | . . . . 5 | |
13 | erth.2 | . . . . . 6 | |
14 | 13 | adantr 274 | . . . . 5 |
15 | elecg 6521 | . . . . 5 | |
16 | 12, 14, 15 | sylancr 411 | . . . 4 |
17 | errel 6492 | . . . . . . 7 | |
18 | 2, 17 | syl 14 | . . . . . 6 |
19 | brrelex2 4630 | . . . . . 6 | |
20 | 18, 19 | sylan 281 | . . . . 5 |
21 | elecg 6521 | . . . . 5 | |
22 | 12, 20, 21 | sylancr 411 | . . . 4 |
23 | 11, 16, 22 | 3bitr4d 219 | . . 3 |
24 | 23 | eqrdv 2155 | . 2 |
25 | 2 | adantr 274 | . . 3 |
26 | 2, 13 | erref 6503 | . . . . . . 7 |
27 | 26 | adantr 274 | . . . . . 6 |
28 | 13 | adantr 274 | . . . . . . 7 |
29 | elecg 6521 | . . . . . . 7 | |
30 | 28, 28, 29 | syl2anc 409 | . . . . . 6 |
31 | 27, 30 | mpbird 166 | . . . . 5 |
32 | simpr 109 | . . . . 5 | |
33 | 31, 32 | eleqtrd 2236 | . . . 4 |
34 | 25, 32 | ereldm 6526 | . . . . . 6 |
35 | 28, 34 | mpbid 146 | . . . . 5 |
36 | elecg 6521 | . . . . 5 | |
37 | 28, 35, 36 | syl2anc 409 | . . . 4 |
38 | 33, 37 | mpbid 146 | . . 3 |
39 | 25, 38 | ersym 6495 | . 2 |
40 | 24, 39 | impbida 586 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cvv 2712 class class class wbr 3967 wrel 4594 wer 6480 cec 6481 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-er 6483 df-ec 6485 |
This theorem is referenced by: erth2 6528 erthi 6529 qliftfun 6565 eroveu 6574 th3qlem1 6585 enqeceq 7282 enq0eceq 7360 nnnq0lem1 7369 enreceq 7659 prsrlem1 7665 |
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