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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iundifdif | Structured version Unicode version |
Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 26056 (Contributed by Thierry Arnoux, 4-Sep-2016.) |
Ref | Expression |
---|---|
iundifdif.o |
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iundifdif.2 |
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Ref | Expression |
---|---|
iundifdif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iundif2 4338 |
. . . 4
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2 | intiin 4325 |
. . . . 5
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3 | 2 | difeq2i 3572 |
. . . 4
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4 | 1, 3 | eqtr4i 2483 |
. . 3
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5 | 4 | difeq2i 3572 |
. 2
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6 | iundifdif.2 |
. . . . 5
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7 | 6 | jctl 541 |
. . . 4
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8 | intssuni2 4254 |
. . . 4
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9 | unipw 4643 |
. . . . . 6
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10 | 9 | sseq2i 3482 |
. . . . 5
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11 | 10 | biimpi 194 |
. . . 4
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12 | 7, 8, 11 | 3syl 20 |
. . 3
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13 | dfss4 3685 |
. . 3
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14 | 12, 13 | sylib 196 |
. 2
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15 | 5, 14 | syl5req 2505 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pr 4632 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-pw 3963 df-sn 3979 df-pr 3981 df-uni 4193 df-int 4230 df-iun 4274 df-iin 4275 |
This theorem is referenced by: (None) |
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