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Mirrors > Home > MPE Home > Th. List > disjen | Structured version Visualization version Unicode version |
Description: A stronger form of pwuninel 7401. We can use pwuninel 7401, 2pwuninel 8115 to create one or two sets disjoint from a given set , but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set we can construct a set that is equinumerous to it and disjoint from . (Contributed by Mario Carneiro, 7-Feb-2015.) |
Ref | Expression |
---|---|
disjen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 7205 | . . . . . . . 8 | |
2 | 1 | ad2antll 765 | . . . . . . 7 |
3 | simprl 794 | . . . . . . 7 | |
4 | 2, 3 | eqeltrrd 2702 | . . . . . 6 |
5 | fvex 6201 | . . . . . . 7 | |
6 | fvex 6201 | . . . . . . 7 | |
7 | 5, 6 | opelrn 5357 | . . . . . 6 |
8 | 4, 7 | syl 17 | . . . . 5 |
9 | pwuninel 7401 | . . . . . 6 | |
10 | xp2nd 7199 | . . . . . . . . 9 | |
11 | 10 | ad2antll 765 | . . . . . . . 8 |
12 | elsni 4194 | . . . . . . . 8 | |
13 | 11, 12 | syl 17 | . . . . . . 7 |
14 | 13 | eleq1d 2686 | . . . . . 6 |
15 | 9, 14 | mtbiri 317 | . . . . 5 |
16 | 8, 15 | pm2.65da 600 | . . . 4 |
17 | elin 3796 | . . . 4 | |
18 | 16, 17 | sylnibr 319 | . . 3 |
19 | 18 | eq0rdv 3979 | . 2 |
20 | simpr 477 | . . 3 | |
21 | rnexg 7098 | . . . . 5 | |
22 | 21 | adantr 481 | . . . 4 |
23 | uniexg 6955 | . . . 4 | |
24 | pwexg 4850 | . . . 4 | |
25 | 22, 23, 24 | 3syl 18 | . . 3 |
26 | xpsneng 8045 | . . 3 | |
27 | 20, 25, 26 | syl2anc 693 | . 2 |
28 | 19, 27 | jca 554 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cin 3573 c0 3915 cpw 4158 csn 4177 cop 4183 cuni 4436 class class class wbr 4653 cxp 5112 crn 5115 cfv 5888 c1st 7166 c2nd 7167 cen 7952 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1st 7168 df-2nd 7169 df-en 7956 |
This theorem is referenced by: disjenex 8118 domss2 8119 |
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