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Mirrors > Home > ILE Home > Th. List > unssin | Unicode version |
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
unssin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oranim 776 | . . . . 5 | |
2 | eldifn 3250 | . . . . . 6 | |
3 | eldifn 3250 | . . . . . 6 | |
4 | 2, 3 | anim12i 336 | . . . . 5 |
5 | 1, 4 | nsyl 623 | . . . 4 |
6 | elin 3310 | . . . 4 | |
7 | 5, 6 | sylnibr 672 | . . 3 |
8 | elun 3268 | . . 3 | |
9 | vex 2733 | . . . 4 | |
10 | eldif 3130 | . . . 4 | |
11 | 9, 10 | mpbiran 935 | . . 3 |
12 | 7, 8, 11 | 3imtr4i 200 | . 2 |
13 | 12 | ssriv 3151 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 703 wcel 2141 cvv 2730 cdif 3118 cun 3119 cin 3120 wss 3121 |
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-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 |
This theorem is referenced by: (None) |
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