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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 6411 | . . . . . . . 8 |
4 | 3 | biimpa 294 | . . . . . . 7 |
5 | 1, 4 | jca 304 | . . . . . 6 |
6 | 2 | ertr 6412 | . . . . . . 7 |
7 | 6 | impl 377 | . . . . . 6 |
8 | 5, 7 | sylan 281 | . . . . 5 |
9 | 2 | ertr 6412 | . . . . . 6 |
10 | 9 | impl 377 | . . . . 5 |
11 | 8, 10 | impbida 570 | . . . 4 |
12 | vex 2663 | . . . . 5 | |
13 | erth.2 | . . . . . 6 | |
14 | 13 | adantr 274 | . . . . 5 |
15 | elecg 6435 | . . . . 5 | |
16 | 12, 14, 15 | sylancr 410 | . . . 4 |
17 | errel 6406 | . . . . . . 7 | |
18 | 2, 17 | syl 14 | . . . . . 6 |
19 | brrelex2 4550 | . . . . . 6 | |
20 | 18, 19 | sylan 281 | . . . . 5 |
21 | elecg 6435 | . . . . 5 | |
22 | 12, 20, 21 | sylancr 410 | . . . 4 |
23 | 11, 16, 22 | 3bitr4d 219 | . . 3 |
24 | 23 | eqrdv 2115 | . 2 |
25 | 2 | adantr 274 | . . 3 |
26 | 2, 13 | erref 6417 | . . . . . . 7 |
27 | 26 | adantr 274 | . . . . . 6 |
28 | 13 | adantr 274 | . . . . . . 7 |
29 | elecg 6435 | . . . . . . 7 | |
30 | 28, 28, 29 | syl2anc 408 | . . . . . 6 |
31 | 27, 30 | mpbird 166 | . . . . 5 |
32 | simpr 109 | . . . . 5 | |
33 | 31, 32 | eleqtrd 2196 | . . . 4 |
34 | 25, 32 | ereldm 6440 | . . . . . 6 |
35 | 28, 34 | mpbid 146 | . . . . 5 |
36 | elecg 6435 | . . . . 5 | |
37 | 28, 35, 36 | syl2anc 408 | . . . 4 |
38 | 33, 37 | mpbid 146 | . . 3 |
39 | 25, 38 | ersym 6409 | . 2 |
40 | 24, 39 | impbida 570 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 class class class wbr 3899 wrel 4514 wer 6394 cec 6395 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-er 6397 df-ec 6399 |
This theorem is referenced by: erth2 6442 erthi 6443 qliftfun 6479 eroveu 6488 th3qlem1 6499 enqeceq 7135 enq0eceq 7213 nnnq0lem1 7222 enreceq 7512 prsrlem1 7518 |
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