Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3637 | . . . . 5 | |
2 | 1 | eqcoms 2160 | . . . 4 |
3 | enpr1g 6743 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | prexg 4171 | . . . . . . 7 | |
6 | eqeng 6711 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 |
8 | entr 6729 | . . . . . . . . 9 | |
9 | 1nen2 6806 | . . . . . . . . . . 11 | |
10 | ensym 6726 | . . . . . . . . . . . 12 | |
11 | entr 6729 | . . . . . . . . . . . . 13 | |
12 | 11 | ex 114 | . . . . . . . . . . . 12 |
13 | 10, 12 | syl 14 | . . . . . . . . . . 11 |
14 | 9, 13 | mtoi 654 | . . . . . . . . . 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 2384 | . 2 |
23 | pr2nelem 7126 | . . 3 | |
24 | 23 | 3expia 1187 | . 2 |
25 | 22, 24 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wne 2327 cvv 2712 cpr 3561 class class class wbr 3965 c1o 6356 c2o 6357 cen 6683 |
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 604 ax-in2 605 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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 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-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-2o 6364 df-er 6480 df-en 6686 |
This theorem is referenced by: exmidonfinlem 7128 pw1dom2 7162 isprm2lem 11993 |
Copyright terms: Public domain | W3C validator |