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Mirrors > Home > ILE Home > Th. List > inssun | Unicode version |
Description: Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
inssun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.1 743 | . . . . 5 | |
2 | eldifn 3194 | . . . . . 6 | |
3 | eldifn 3194 | . . . . . 6 | |
4 | 2, 3 | orim12i 748 | . . . . 5 |
5 | 1, 4 | nsyl 617 | . . . 4 |
6 | elun 3212 | . . . 4 | |
7 | 5, 6 | sylnibr 666 | . . 3 |
8 | elin 3254 | . . 3 | |
9 | vex 2684 | . . . 4 | |
10 | eldif 3075 | . . . 4 | |
11 | 9, 10 | mpbiran 924 | . . 3 |
12 | 7, 8, 11 | 3imtr4i 200 | . 2 |
13 | 12 | ssriv 3096 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 697 wcel 1480 cvv 2681 cdif 3063 cun 3064 cin 3065 wss 3066 |
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 603 ax-in2 604 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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 |
This theorem is referenced by: (None) |
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