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Mirrors > Home > ILE Home > Th. List > qsel | Unicode version |
Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
qsel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . 3 | |
2 | eleq2 2201 | . . . 4 | |
3 | eqeq1 2144 | . . . 4 | |
4 | 2, 3 | imbi12d 233 | . . 3 |
5 | vex 2684 | . . . . . 6 | |
6 | elecg 6460 | . . . . . 6 | |
7 | 5, 6 | mpan2 421 | . . . . 5 |
8 | 7 | ibi 175 | . . . 4 |
9 | simpll 518 | . . . . . 6 | |
10 | simpr 109 | . . . . . 6 | |
11 | 9, 10 | erthi 6468 | . . . . 5 |
12 | 11 | ex 114 | . . . 4 |
13 | 8, 12 | syl5 32 | . . 3 |
14 | 1, 4, 13 | ectocld 6488 | . 2 |
15 | 14 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cvv 2681 class class class wbr 3924 wer 6419 cec 6420 cqs 6421 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-er 6422 df-ec 6424 df-qs 6428 |
This theorem is referenced by: (None) |
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