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 13193 | . . . . 5 |
3 | 2 | adantr 274 | . . . 4 |
4 | pw1dom2 13190 | . . . . . 6 | |
5 | iunxpconst 4599 | . . . . . . . . . . . 12 | |
6 | df1o2 6326 | . . . . . . . . . . . . 13 | |
7 | 6 | xpeq2i 4560 | . . . . . . . . . . . 12 |
8 | 1, 5, 7 | 3eqtri 2164 | . . . . . . . . . . 11 |
9 | peano1 4508 | . . . . . . . . . . . 12 | |
10 | xpsneng 6716 | . . . . . . . . . . . 12 | |
11 | 9, 10 | mpan2 421 | . . . . . . . . . . 11 |
12 | 8, 11 | eqbrtrid 3963 | . . . . . . . . . 10 |
13 | 12 | ad2antrr 479 | . . . . . . . . 9 |
14 | 13 | ensymd 6677 | . . . . . . . 8 |
15 | relen 6638 | . . . . . . . . . 10 | |
16 | brrelex1 4578 | . . . . . . . . . 10 | |
17 | 15, 13, 16 | sylancr 410 | . . . . . . . . 9 |
18 | simplr 519 | . . . . . . . . . 10 | |
19 | simpr 109 | . . . . . . . . . 10 | |
20 | dff1o5 5376 | . . . . . . . . . 10 | |
21 | 18, 19, 20 | sylanbrc 413 | . . . . . . . . 9 |
22 | f1oeng 6651 | . . . . . . . . 9 | |
23 | 17, 21, 22 | syl2anc 408 | . . . . . . . 8 |
24 | entr 6678 | . . . . . . . 8 | |
25 | 14, 23, 24 | syl2anc 408 | . . . . . . 7 |
26 | 25 | ensymd 6677 | . . . . . 6 |
27 | domentr 6685 | . . . . . 6 | |
28 | 4, 26, 27 | sylancr 410 | . . . . 5 |
29 | 2onn 6417 | . . . . . . 7 | |
30 | nndomo 6758 | . . . . . . 7 | |
31 | 29, 30 | mpan 420 | . . . . . 6 |
32 | 31 | ad2antrr 479 | . . . . 5 |
33 | 28, 32 | mpbid 146 | . . . 4 |
34 | 3, 33 | eqssd 3114 | . . 3 |
35 | 26, 34 | breqtrd 3954 | . . . 4 |
36 | exmidpw 6802 | . . . 4 EXMID | |
37 | 35, 36 | sylibr 133 | . . 3 EXMID |
38 | 34, 37 | jca 304 | . 2 EXMID |
39 | simplr 519 | . . . . 5 EXMID | |
40 | 12 | ad2antrr 479 | . . . . . . . 8 EXMID |
41 | simprl 520 | . . . . . . . 8 EXMID | |
42 | 40, 41 | breqtrd 3954 | . . . . . . 7 EXMID |
43 | simprr 521 | . . . . . . . . 9 EXMID EXMID | |
44 | 43, 36 | sylib 121 | . . . . . . . 8 EXMID |
45 | 44 | ensymd 6677 | . . . . . . 7 EXMID |
46 | entr 6678 | . . . . . . 7 | |
47 | 42, 45, 46 | syl2anc 408 | . . . . . 6 EXMID |
48 | nnfi 6766 | . . . . . . . 8 | |
49 | 29, 48 | mp1i 10 | . . . . . . 7 EXMID |
50 | enfi 6767 | . . . . . . . 8 | |
51 | 44, 50 | syl 14 | . . . . . . 7 EXMID |
52 | 49, 51 | mpbird 166 | . . . . . 6 EXMID |
53 | f1finf1o 6835 | . . . . . 6 | |
54 | 47, 52, 53 | syl2anc 408 | . . . . 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 585 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 wss 3071 c0 3363 cpw 3510 csn 3527 ciun 3813 class class class wbr 3929 EXMIDwem 4118 com 4504 cxp 4537 crn 4540 wrel 4544 wf1 5120 wf1o 5122 c1o 6306 c2o 6307 cen 6632 cdom 6633 cfn 6634 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-exmid 4119 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-2o 6314 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 |
This theorem is referenced by: (None) |
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