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Mirrors > Home > ILE Home > Th. List > pr2ne | Unicode version |
Description: If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
Ref | Expression |
---|---|
pr2ne |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3661 | . . . . 5 | |
2 | 1 | eqcoms 2173 | . . . 4 |
3 | enpr1g 6776 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | prexg 4196 | . . . . . . 7 | |
6 | eqeng 6744 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 |
8 | entr 6762 | . . . . . . . . 9 | |
9 | 1nen2 6839 | . . . . . . . . . . 11 | |
10 | ensym 6759 | . . . . . . . . . . . 12 | |
11 | entr 6762 | . . . . . . . . . . . . 13 | |
12 | 11 | ex 114 | . . . . . . . . . . . 12 |
13 | 10, 12 | syl 14 | . . . . . . . . . . 11 |
14 | 9, 13 | mtoi 659 | . . . . . . . . . 10 |
15 | 14 | a1d 22 | . . . . . . . . 9 |
16 | 8, 15 | syl 14 | . . . . . . . 8 |
17 | 16 | ex 114 | . . . . . . 7 |
18 | 17 | com3r 79 | . . . . . 6 |
19 | 7, 18 | syld 45 | . . . . 5 |
20 | 4, 19 | mpid 42 | . . . 4 |
21 | 2, 20 | syl5 32 | . . 3 |
22 | 21 | necon2ad 2397 | . 2 |
23 | pr2nelem 7168 | . . 3 | |
24 | 23 | 3expia 1200 | . 2 |
25 | 22, 24 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wne 2340 cvv 2730 cpr 3584 class class class wbr 3989 c1o 6388 c2o 6389 cen 6716 |
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-in1 609 ax-in2 610 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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 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-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 |
This theorem is referenced by: exmidonfinlem 7170 pw1dom2 7204 isprm2lem 12070 |
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