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Mirrors > Home > ILE Home > Th. List > difprsn1 | Unicode version |
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
difprsn1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necom 2369 | . 2 | |
2 | disjsn2 3556 | . . . 4 | |
3 | disj3 3385 | . . . 4 | |
4 | 2, 3 | sylib 121 | . . 3 |
5 | df-pr 3504 | . . . . . 6 | |
6 | 5 | equncomi 3192 | . . . . 5 |
7 | 6 | difeq1i 3160 | . . . 4 |
8 | difun2 3412 | . . . 4 | |
9 | 7, 8 | eqtri 2138 | . . 3 |
10 | 4, 9 | syl6reqr 2169 | . 2 |
11 | 1, 10 | sylbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wne 2285 cdif 3038 cun 3039 cin 3040 c0 3333 csn 3497 cpr 3498 |
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 588 ax-in2 589 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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 |
This theorem is referenced by: difprsn2 3630 |
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