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Mirrors > Home > ILE Home > Th. List > pm54.43 | Unicode version |
Description: Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
Ref | Expression |
---|---|
pm54.43 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 6031 | . . . . . . . 8 | |
2 | 1 | elexi 2611 | . . . . . . 7 |
3 | 2 | ensn1 6299 | . . . . . 6 |
4 | 3 | ensymi 6285 | . . . . 5 |
5 | entr 6287 | . . . . 5 | |
6 | 4, 5 | mpan2 415 | . . . 4 |
7 | 1 | onirri 4286 | . . . . . . 7 |
8 | disjsn 3454 | . . . . . . 7 | |
9 | 7, 8 | mpbir 144 | . . . . . 6 |
10 | unen 6316 | . . . . . 6 | |
11 | 9, 10 | mpanr2 428 | . . . . 5 |
12 | 11 | ex 113 | . . . 4 |
13 | 6, 12 | sylan2 280 | . . 3 |
14 | df-2o 6025 | . . . . 5 | |
15 | df-suc 4126 | . . . . 5 | |
16 | 14, 15 | eqtri 2101 | . . . 4 |
17 | 16 | breq2i 3793 | . . 3 |
18 | 13, 17 | syl6ibr 160 | . 2 |
19 | en1 6302 | . . 3 | |
20 | en1 6302 | . . 3 | |
21 | 1nen2 6347 | . . . . . . . . . . . . 13 | |
22 | 21 | a1i 9 | . . . . . . . . . . . 12 |
23 | unidm 3115 | . . . . . . . . . . . . . . . 16 | |
24 | sneq 3409 | . . . . . . . . . . . . . . . . 17 | |
25 | 24 | uneq2d 3126 | . . . . . . . . . . . . . . . 16 |
26 | 23, 25 | syl5reqr 2128 | . . . . . . . . . . . . . . 15 |
27 | vex 2604 | . . . . . . . . . . . . . . . 16 | |
28 | 27 | ensn1 6299 | . . . . . . . . . . . . . . 15 |
29 | 26, 28 | syl6eqbr 3822 | . . . . . . . . . . . . . 14 |
30 | 29 | ensymd 6286 | . . . . . . . . . . . . 13 |
31 | entr 6287 | . . . . . . . . . . . . 13 | |
32 | 30, 31 | sylan 277 | . . . . . . . . . . . 12 |
33 | 22, 32 | mtand 623 | . . . . . . . . . . 11 |
34 | 33 | necon2ai 2299 | . . . . . . . . . 10 |
35 | disjsn2 3455 | . . . . . . . . . 10 | |
36 | 34, 35 | syl 14 | . . . . . . . . 9 |
37 | 36 | a1i 9 | . . . . . . . 8 |
38 | uneq12 3121 | . . . . . . . . 9 | |
39 | 38 | breq1d 3795 | . . . . . . . 8 |
40 | ineq12 3162 | . . . . . . . . 9 | |
41 | 40 | eqeq1d 2089 | . . . . . . . 8 |
42 | 37, 39, 41 | 3imtr4d 201 | . . . . . . 7 |
43 | 42 | ex 113 | . . . . . 6 |
44 | 43 | exlimdv 1740 | . . . . 5 |
45 | 44 | exlimiv 1529 | . . . 4 |
46 | 45 | imp 122 | . . 3 |
47 | 19, 20, 46 | syl2anb 285 | . 2 |
48 | 18, 47 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wne 2245 cun 2971 cin 2972 c0 3251 csn 3398 class class class wbr 3785 con0 4118 csuc 4120 c1o 6017 c2o 6018 cen 6242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-2o 6025 df-er 6129 df-en 6245 |
This theorem is referenced by: pr2nelem 6460 |
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