Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iundifdif | Structured version Visualization version Unicode version |
Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 29380. (Contributed by Thierry Arnoux, 4-Sep-2016.) |
Ref | Expression |
---|---|
iundifdif.o | |
iundifdif.2 |
Ref | Expression |
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iundifdif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iundif2 4587 | . . . 4 | |
2 | intiin 4574 | . . . . 5 | |
3 | 2 | difeq2i 3725 | . . . 4 |
4 | 1, 3 | eqtr4i 2647 | . . 3 |
5 | 4 | difeq2i 3725 | . 2 |
6 | iundifdif.2 | . . . . 5 | |
7 | 6 | jctl 564 | . . . 4 |
8 | intssuni2 4502 | . . . 4 | |
9 | unipw 4918 | . . . . . 6 | |
10 | 9 | sseq2i 3630 | . . . . 5 |
11 | 10 | biimpi 206 | . . . 4 |
12 | 7, 8, 11 | 3syl 18 | . . 3 |
13 | dfss4 3858 | . . 3 | |
14 | 12, 13 | sylib 208 | . 2 |
15 | 5, 14 | syl5req 2669 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 cdif 3571 wss 3574 c0 3915 cpw 4158 cuni 4436 cint 4475 ciun 4520 ciin 4521 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 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-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 |
This theorem is referenced by: (None) |
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