Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
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 13712 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | pw1dom2 7174 | . . . . . 6 | |
5 | iunxpconst 4658 | . . . . . . . . . . . 12 | |
6 | df1o2 6388 | . . . . . . . . . . . . 13 | |
7 | 6 | xpeq2i 4619 | . . . . . . . . . . . 12 |
8 | 1, 5, 7 | 3eqtri 2189 | . . . . . . . . . . 11 |
9 | peano1 4565 | . . . . . . . . . . . 12 | |
10 | xpsneng 6779 | . . . . . . . . . . . 12 | |
11 | 9, 10 | mpan2 422 | . . . . . . . . . . 11 |
12 | 8, 11 | eqbrtrid 4011 | . . . . . . . . . 10 |
13 | 12 | ad2antrr 480 | . . . . . . . . 9 |
14 | 13 | ensymd 6740 | . . . . . . . 8 |
15 | relen 6701 | . . . . . . . . . 10 | |
16 | brrelex1 4637 | . . . . . . . . . 10 | |
17 | 15, 13, 16 | sylancr 411 | . . . . . . . . 9 |
18 | simplr 520 | . . . . . . . . . 10 | |
19 | simpr 109 | . . . . . . . . . 10 | |
20 | dff1o5 5435 | . . . . . . . . . 10 | |
21 | 18, 19, 20 | sylanbrc 414 | . . . . . . . . 9 |
22 | f1oeng 6714 | . . . . . . . . 9 | |
23 | 17, 21, 22 | syl2anc 409 | . . . . . . . 8 |
24 | entr 6741 | . . . . . . . 8 | |
25 | 14, 23, 24 | syl2anc 409 | . . . . . . 7 |
26 | 25 | ensymd 6740 | . . . . . 6 |
27 | domentr 6748 | . . . . . 6 | |
28 | 4, 26, 27 | sylancr 411 | . . . . 5 |
29 | 2onn 6480 | . . . . . . 7 | |
30 | nndomo 6821 | . . . . . . 7 | |
31 | 29, 30 | mpan 421 | . . . . . 6 |
32 | 31 | ad2antrr 480 | . . . . 5 |
33 | 28, 32 | mpbid 146 | . . . 4 |
34 | 3, 33 | eqssd 3154 | . . 3 |
35 | 26, 34 | breqtrd 4002 | . . . 4 |
36 | exmidpw 6865 | . . . 4 EXMID | |
37 | 35, 36 | sylibr 133 | . . 3 EXMID |
38 | 34, 37 | jca 304 | . 2 EXMID |
39 | simplr 520 | . . . . 5 EXMID | |
40 | 12 | ad2antrr 480 | . . . . . . . 8 EXMID |
41 | simprl 521 | . . . . . . . 8 EXMID | |
42 | 40, 41 | breqtrd 4002 | . . . . . . 7 EXMID |
43 | simprr 522 | . . . . . . . . 9 EXMID EXMID | |
44 | 43, 36 | sylib 121 | . . . . . . . 8 EXMID |
45 | 44 | ensymd 6740 | . . . . . . 7 EXMID |
46 | entr 6741 | . . . . . . 7 | |
47 | 42, 45, 46 | syl2anc 409 | . . . . . 6 EXMID |
48 | nnfi 6829 | . . . . . . . 8 | |
49 | 29, 48 | mp1i 10 | . . . . . . 7 EXMID |
50 | enfi 6830 | . . . . . . . 8 | |
51 | 44, 50 | syl 14 | . . . . . . 7 EXMID |
52 | 49, 51 | mpbird 166 | . . . . . 6 EXMID |
53 | f1finf1o 6903 | . . . . . 6 | |
54 | 47, 52, 53 | syl2anc 409 | . . . . 5 EXMID |
55 | 39, 54 | mpbid 146 | . . . 4 EXMID |
56 | 55, 20 | sylib 121 | . . 3 EXMID |
57 | 56 | simprd 113 | . 2 EXMID |
58 | 38, 57 | impbida 586 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 wss 3111 c0 3404 cpw 3553 csn 3570 ciun 3860 class class class wbr 3976 EXMIDwem 4167 com 4561 cxp 4596 crn 4599 wrel 4603 wf1 5179 wf1o 5181 c1o 6368 c2o 6369 cen 6695 cdom 6696 cfn 6697 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-exmid 4168 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1o 6375 df-2o 6376 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |