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Mirrors > Home > MPE Home > Th. List > unissint | Structured version Visualization version 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 4514). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
unissint |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . . 5 | |
2 | df-ne 2795 | . . . . . . 7 | |
3 | intssuni 4499 | . . . . . . 7 | |
4 | 2, 3 | sylbir 225 | . . . . . 6 |
5 | 4 | adantl 482 | . . . . 5 |
6 | 1, 5 | eqssd 3620 | . . . 4 |
7 | 6 | ex 450 | . . 3 |
8 | 7 | orrd 393 | . 2 |
9 | ssv 3625 | . . . . 5 | |
10 | int0 4490 | . . . . 5 | |
11 | 9, 10 | sseqtr4i 3638 | . . . 4 |
12 | inteq 4478 | . . . 4 | |
13 | 11, 12 | syl5sseqr 3654 | . . 3 |
14 | eqimss 3657 | . . 3 | |
15 | 13, 14 | jaoi 394 | . 2 |
16 | 8, 15 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wo 383 wa 384 wceq 1483 wne 2794 cvv 3200 wss 3574 c0 3915 cuni 4436 cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-uni 4437 df-int 4476 |
This theorem is referenced by: (None) |
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