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Mirrors > Home > MPE Home > Th. List > uniinqs | Structured version Unicode version |
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4214. (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 4214 |
. . 3
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2 | 1 | a1i 11 |
. 2
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3 | eluni2 4198 |
. . . . . 6
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4 | eluni2 4198 |
. . . . . 6
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5 | 3, 4 | anbi12i 697 |
. . . . 5
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6 | elin 3642 |
. . . . 5
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7 | reeanv 2988 |
. . . . 5
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8 | 5, 6, 7 | 3bitr4i 277 |
. . . 4
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9 | simp3l 1016 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | simp2l 1014 |
. . . . . . . 8
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11 | inelcm 3836 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 11 | 3ad2ant3 1011 |
. . . . . . . . . 10
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13 | uniinqs.1 |
. . . . . . . . . . . . . 14
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14 | 13 | a1i 11 |
. . . . . . . . . . . . 13
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15 | simp1l 1012 |
. . . . . . . . . . . . . 14
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16 | 15, 10 | sseldd 3460 |
. . . . . . . . . . . . 13
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17 | simp1r 1013 |
. . . . . . . . . . . . . 14
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18 | simp2r 1015 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sseldd 3460 |
. . . . . . . . . . . . 13
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20 | 14, 16, 19 | qsdisj 7282 |
. . . . . . . . . . . 12
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21 | 20 | ord 377 |
. . . . . . . . . . 11
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22 | 21 | necon1ad 2665 |
. . . . . . . . . 10
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23 | 12, 22 | mpd 15 |
. . . . . . . . 9
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24 | 23, 18 | eqeltrd 2540 |
. . . . . . . 8
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25 | 10, 24 | elind 3643 |
. . . . . . 7
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26 | elunii 4199 |
. . . . . . 7
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27 | 9, 25, 26 | syl2anc 661 |
. . . . . 6
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28 | 27 | 3expia 1190 |
. . . . 5
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29 | 28 | rexlimdvva 2948 |
. . . 4
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30 | 8, 29 | syl5bi 217 |
. . 3
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31 | 30 | ssrdv 3465 |
. 2
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32 | 2, 31 | eqssd 3476 |
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 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pr 4634 |
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-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-er 7206 df-ec 7208 df-qs 7212 |
This theorem is referenced by: (None) |
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