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Mirrors > Home > MPE Home > Th. List > endisj | Structured version Visualization version Unicode version |
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) |
Ref | Expression |
---|---|
endisj.1 | |
endisj.2 |
Ref | Expression |
---|---|
endisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endisj.1 | . . . 4 | |
2 | 0ex 4790 | . . . 4 | |
3 | 1, 2 | xpsnen 8044 | . . 3 |
4 | endisj.2 | . . . 4 | |
5 | 1on 7567 | . . . . 5 | |
6 | 5 | elexi 3213 | . . . 4 |
7 | 4, 6 | xpsnen 8044 | . . 3 |
8 | 3, 7 | pm3.2i 471 | . 2 |
9 | xp01disj 7576 | . 2 | |
10 | p0ex 4853 | . . . 4 | |
11 | 1, 10 | xpex 6962 | . . 3 |
12 | snex 4908 | . . . 4 | |
13 | 4, 12 | xpex 6962 | . . 3 |
14 | breq1 4656 | . . . . 5 | |
15 | breq1 4656 | . . . . 5 | |
16 | 14, 15 | bi2anan9 917 | . . . 4 |
17 | ineq12 3809 | . . . . 5 | |
18 | 17 | eqeq1d 2624 | . . . 4 |
19 | 16, 18 | anbi12d 747 | . . 3 |
20 | 11, 13, 19 | spc2ev 3301 | . 2 |
21 | 8, 9, 20 | mp2an 708 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 cin 3573 c0 3915 csn 4177 class class class wbr 4653 cxp 5112 con0 5723 c1o 7553 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-3or 1038 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-ne 2795 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-ord 5726 df-on 5727 df-suc 5729 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-1o 7560 df-en 7956 |
This theorem is referenced by: (None) |
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