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Mirrors > Home > NFE Home > Th. List > unissint | Unicode version |
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3963). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
unissint |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 443 | . . . . 5 | |
2 | df-ne 2518 | . . . . . . 7 | |
3 | intssuni 3948 | . . . . . . 7 | |
4 | 2, 3 | sylbir 204 | . . . . . 6 |
5 | 4 | adantl 452 | . . . . 5 |
6 | 1, 5 | eqssd 3289 | . . . 4 |
7 | 6 | ex 423 | . . 3 |
8 | 7 | orrd 367 | . 2 |
9 | ssv 3291 | . . . . 5 | |
10 | int0 3940 | . . . . 5 | |
11 | 9, 10 | sseqtr4i 3304 | . . . 4 |
12 | inteq 3929 | . . . 4 | |
13 | 11, 12 | syl5sseqr 3320 | . . 3 |
14 | eqimss 3323 | . . 3 | |
15 | 13, 14 | jaoi 368 | . 2 |
16 | 8, 15 | impbii 180 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 176 wo 357 wa 358 wceq 1642 wne 2516 cvv 2859 wss 3257 c0 3550 cuni 3891 cint 3926 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-uni 3892 df-int 3927 |
This theorem is referenced by: (None) |
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