Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > pwf1oexmid | Unicode version |
Description: An exercise related to copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.) |
Ref | Expression |
---|---|
pwle2.t |
Ref | Expression |
---|---|
pwf1oexmid | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwle2.t | . . . . . 6 | |
2 | 1 | pwle2 14308 | . . . . 5 |
3 | 2 | adantr 276 | . . . 4 |
4 | pw1dom2 7216 | . . . . . 6 | |
5 | iunxpconst 4680 | . . . . . . . . . . . 12 | |
6 | df1o2 6420 | . . . . . . . . . . . . 13 | |
7 | 6 | xpeq2i 4641 | . . . . . . . . . . . 12 |
8 | 1, 5, 7 | 3eqtri 2200 | . . . . . . . . . . 11 |
9 | peano1 4587 | . . . . . . . . . . . 12 | |
10 | xpsneng 6812 | . . . . . . . . . . . 12 | |
11 | 9, 10 | mpan2 425 | . . . . . . . . . . 11 |
12 | 8, 11 | eqbrtrid 4033 | . . . . . . . . . 10 |
13 | 12 | ad2antrr 488 | . . . . . . . . 9 |
14 | 13 | ensymd 6773 | . . . . . . . 8 |
15 | relen 6734 | . . . . . . . . . 10 | |
16 | brrelex1 4659 | . . . . . . . . . 10 | |
17 | 15, 13, 16 | sylancr 414 | . . . . . . . . 9 |
18 | simplr 528 | . . . . . . . . . 10 | |
19 | simpr 110 | . . . . . . . . . 10 | |
20 | dff1o5 5462 | . . . . . . . . . 10 | |
21 | 18, 19, 20 | sylanbrc 417 | . . . . . . . . 9 |
22 | f1oeng 6747 | . . . . . . . . 9 | |
23 | 17, 21, 22 | syl2anc 411 | . . . . . . . 8 |
24 | entr 6774 | . . . . . . . 8 | |
25 | 14, 23, 24 | syl2anc 411 | . . . . . . 7 |
26 | 25 | ensymd 6773 | . . . . . 6 |
27 | domentr 6781 | . . . . . 6 | |
28 | 4, 26, 27 | sylancr 414 | . . . . 5 |
29 | 2onn 6512 | . . . . . . 7 | |
30 | nndomo 6854 | . . . . . . 7 | |
31 | 29, 30 | mpan 424 | . . . . . 6 |
32 | 31 | ad2antrr 488 | . . . . 5 |
33 | 28, 32 | mpbid 147 | . . . 4 |
34 | 3, 33 | eqssd 3170 | . . 3 |
35 | 26, 34 | breqtrd 4024 | . . . 4 |
36 | exmidpw 6898 | . . . 4 EXMID | |
37 | 35, 36 | sylibr 134 | . . 3 EXMID |
38 | 34, 37 | jca 306 | . 2 EXMID |
39 | simplr 528 | . . . . 5 EXMID | |
40 | 12 | ad2antrr 488 | . . . . . . . 8 EXMID |
41 | simprl 529 | . . . . . . . 8 EXMID | |
42 | 40, 41 | breqtrd 4024 | . . . . . . 7 EXMID |
43 | simprr 531 | . . . . . . . . 9 EXMID EXMID | |
44 | 43, 36 | sylib 122 | . . . . . . . 8 EXMID |
45 | 44 | ensymd 6773 | . . . . . . 7 EXMID |
46 | entr 6774 | . . . . . . 7 | |
47 | 42, 45, 46 | syl2anc 411 | . . . . . 6 EXMID |
48 | nnfi 6862 | . . . . . . . 8 | |
49 | 29, 48 | mp1i 10 | . . . . . . 7 EXMID |
50 | enfi 6863 | . . . . . . . 8 | |
51 | 44, 50 | syl 14 | . . . . . . 7 EXMID |
52 | 49, 51 | mpbird 167 | . . . . . 6 EXMID |
53 | f1finf1o 6936 | . . . . . 6 | |
54 | 47, 52, 53 | syl2anc 411 | . . . . 5 EXMID |
55 | 39, 54 | mpbid 147 | . . . 4 EXMID |
56 | 55, 20 | sylib 122 | . . 3 EXMID |
57 | 56 | simprd 114 | . 2 EXMID |
58 | 38, 57 | impbida 596 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 cvv 2735 wss 3127 c0 3420 cpw 3572 csn 3589 ciun 3882 class class class wbr 3998 EXMIDwem 4189 com 4583 cxp 4618 crn 4621 wrel 4625 wf1 5205 wf1o 5207 c1o 6400 c2o 6401 cen 6728 cdom 6729 cfn 6730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-exmid 4190 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1o 6407 df-2o 6408 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 |
This theorem is referenced by: (None) |
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