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Mirrors > Home > ILE Home > Th. List > uneqin | Unicode version |
Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
uneqin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3151 | . . . 4 | |
2 | unss 3250 | . . . . 5 | |
3 | ssin 3298 | . . . . . . 7 | |
4 | sstr 3105 | . . . . . . 7 | |
5 | 3, 4 | sylbir 134 | . . . . . 6 |
6 | ssin 3298 | . . . . . . 7 | |
7 | simpl 108 | . . . . . . 7 | |
8 | 6, 7 | sylbir 134 | . . . . . 6 |
9 | 5, 8 | anim12i 336 | . . . . 5 |
10 | 2, 9 | sylbir 134 | . . . 4 |
11 | 1, 10 | syl 14 | . . 3 |
12 | eqss 3112 | . . 3 | |
13 | 11, 12 | sylibr 133 | . 2 |
14 | unidm 3219 | . . . 4 | |
15 | inidm 3285 | . . . 4 | |
16 | 14, 15 | eqtr4i 2163 | . . 3 |
17 | uneq2 3224 | . . 3 | |
18 | ineq2 3271 | . . 3 | |
19 | 16, 17, 18 | 3eqtr3a 2196 | . 2 |
20 | 13, 19 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 cun 3069 cin 3070 wss 3071 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 |
This theorem is referenced by: (None) |
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