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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 6313 | . . . . . . . 8 | |
2 | 1 | elexi 2693 | . . . . . . 7 |
3 | 2 | ensn1 6683 | . . . . . 6 |
4 | 3 | ensymi 6669 | . . . . 5 |
5 | entr 6671 | . . . . 5 | |
6 | 4, 5 | mpan2 421 | . . . 4 |
7 | 1 | onirri 4453 | . . . . . . 7 |
8 | disjsn 3580 | . . . . . . 7 | |
9 | 7, 8 | mpbir 145 | . . . . . 6 |
10 | unen 6703 | . . . . . 6 | |
11 | 9, 10 | mpanr2 434 | . . . . 5 |
12 | 11 | ex 114 | . . . 4 |
13 | 6, 12 | sylan2 284 | . . 3 |
14 | df-2o 6307 | . . . . 5 | |
15 | df-suc 4288 | . . . . 5 | |
16 | 14, 15 | eqtri 2158 | . . . 4 |
17 | 16 | breq2i 3932 | . . 3 |
18 | 13, 17 | syl6ibr 161 | . 2 |
19 | en1 6686 | . . 3 | |
20 | en1 6686 | . . 3 | |
21 | 1nen2 6748 | . . . . . . . . . . . . 13 | |
22 | 21 | a1i 9 | . . . . . . . . . . . 12 |
23 | unidm 3214 | . . . . . . . . . . . . . . . 16 | |
24 | sneq 3533 | . . . . . . . . . . . . . . . . 17 | |
25 | 24 | uneq2d 3225 | . . . . . . . . . . . . . . . 16 |
26 | 23, 25 | syl5reqr 2185 | . . . . . . . . . . . . . . 15 |
27 | vex 2684 | . . . . . . . . . . . . . . . 16 | |
28 | 27 | ensn1 6683 | . . . . . . . . . . . . . . 15 |
29 | 26, 28 | eqbrtrdi 3962 | . . . . . . . . . . . . . 14 |
30 | 29 | ensymd 6670 | . . . . . . . . . . . . 13 |
31 | entr 6671 | . . . . . . . . . . . . 13 | |
32 | 30, 31 | sylan 281 | . . . . . . . . . . . 12 |
33 | 22, 32 | mtand 654 | . . . . . . . . . . 11 |
34 | 33 | necon2ai 2360 | . . . . . . . . . 10 |
35 | disjsn2 3581 | . . . . . . . . . 10 | |
36 | 34, 35 | syl 14 | . . . . . . . . 9 |
37 | 36 | a1i 9 | . . . . . . . 8 |
38 | uneq12 3220 | . . . . . . . . 9 | |
39 | 38 | breq1d 3934 | . . . . . . . 8 |
40 | ineq12 3267 | . . . . . . . . 9 | |
41 | 40 | eqeq1d 2146 | . . . . . . . 8 |
42 | 37, 39, 41 | 3imtr4d 202 | . . . . . . 7 |
43 | 42 | ex 114 | . . . . . 6 |
44 | 43 | exlimdv 1791 | . . . . 5 |
45 | 44 | exlimiv 1577 | . . . 4 |
46 | 45 | imp 123 | . . 3 |
47 | 19, 20, 46 | syl2anb 289 | . 2 |
48 | 18, 47 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wne 2306 cun 3064 cin 3065 c0 3358 csn 3522 class class class wbr 3924 con0 4280 csuc 4282 c1o 6299 c2o 6300 cen 6625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1o 6306 df-2o 6307 df-er 6422 df-en 6628 |
This theorem is referenced by: pr2nelem 7040 dju1p1e2 7046 |
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