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Mirrors > Home > MPE Home > Th. List > uniinqs | Structured version Visualization version Unicode version |
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4221. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.) |
Ref | Expression |
---|---|
uniinqs.1 |
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Ref | Expression |
---|---|
uniinqs |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniin 4221 |
. . 3
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2 | 1 | a1i 11 |
. 2
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3 | eluni2 4205 |
. . . . . 6
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4 | eluni2 4205 |
. . . . . 6
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5 | 3, 4 | anbi12i 704 |
. . . . 5
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6 | elin 3619 |
. . . . 5
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7 | reeanv 2960 |
. . . . 5
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8 | 5, 6, 7 | 3bitr4i 281 |
. . . 4
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9 | simp3l 1037 |
. . . . . . 7
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10 | simp2l 1035 |
. . . . . . . 8
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11 | inelcm 3821 |
. . . . . . . . . . 11
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12 | 11 | 3ad2ant3 1032 |
. . . . . . . . . 10
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13 | uniinqs.1 |
. . . . . . . . . . . . . 14
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14 | 13 | a1i 11 |
. . . . . . . . . . . . 13
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15 | simp1l 1033 |
. . . . . . . . . . . . . 14
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16 | 15, 10 | sseldd 3435 |
. . . . . . . . . . . . 13
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17 | simp1r 1034 |
. . . . . . . . . . . . . 14
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18 | simp2r 1036 |
. . . . . . . . . . . . . 14
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19 | 17, 18 | sseldd 3435 |
. . . . . . . . . . . . 13
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20 | 14, 16, 19 | qsdisj 7445 |
. . . . . . . . . . . 12
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21 | 20 | ord 379 |
. . . . . . . . . . 11
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22 | 21 | necon1ad 2643 |
. . . . . . . . . 10
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23 | 12, 22 | mpd 15 |
. . . . . . . . 9
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24 | 23, 18 | eqeltrd 2531 |
. . . . . . . 8
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25 | 10, 24 | elind 3620 |
. . . . . . 7
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26 | elunii 4206 |
. . . . . . 7
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27 | 9, 25, 26 | syl2anc 667 |
. . . . . 6
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28 | 27 | 3expia 1211 |
. . . . 5
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29 | 28 | rexlimdvva 2888 |
. . . 4
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30 | 8, 29 | syl5bi 221 |
. . 3
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31 | 30 | ssrdv 3440 |
. 2
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32 | 2, 31 | eqssd 3451 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-er 7368 df-ec 7370 df-qs 7374 |
This theorem is referenced by: (None) |
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