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Mirrors > Home > ILE Home > Th. List > xpsnen | Unicode version |
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
xpsnen.1 | |
xpsnen.2 |
Ref | Expression |
---|---|
xpsnen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsnen.1 | . . 3 | |
2 | xpsnen.2 | . . . 4 | |
3 | 2 | snex 3937 | . . 3 |
4 | 1, 3 | xpex 4453 | . 2 |
5 | elxp 4362 | . . 3 | |
6 | inteq 3618 | . . . . . . . 8 | |
7 | 6 | inteqd 3620 | . . . . . . 7 |
8 | vex 2560 | . . . . . . . 8 | |
9 | vex 2560 | . . . . . . . 8 | |
10 | 8, 9 | op1stb 4209 | . . . . . . 7 |
11 | 7, 10 | syl6eq 2088 | . . . . . 6 |
12 | 11, 8 | syl6eqel 2128 | . . . . 5 |
13 | 12 | adantr 261 | . . . 4 |
14 | 13 | exlimivv 1776 | . . 3 |
15 | 5, 14 | sylbi 114 | . 2 |
16 | 8, 2 | opex 3966 | . . 3 |
17 | 16 | a1i 9 | . 2 |
18 | eqvisset 2565 | . . . . 5 | |
19 | ancom 253 | . . . . . . . . . . 11 | |
20 | anass 381 | . . . . . . . . . . 11 | |
21 | velsn 3392 | . . . . . . . . . . . 12 | |
22 | 21 | anbi1i 431 | . . . . . . . . . . 11 |
23 | 19, 20, 22 | 3bitr3i 199 | . . . . . . . . . 10 |
24 | 23 | exbii 1496 | . . . . . . . . 9 |
25 | opeq2 3550 | . . . . . . . . . . . 12 | |
26 | 25 | eqeq2d 2051 | . . . . . . . . . . 11 |
27 | 26 | anbi1d 438 | . . . . . . . . . 10 |
28 | 2, 27 | ceqsexv 2593 | . . . . . . . . 9 |
29 | inteq 3618 | . . . . . . . . . . . . . 14 | |
30 | 29 | inteqd 3620 | . . . . . . . . . . . . 13 |
31 | 8, 2 | op1stb 4209 | . . . . . . . . . . . . 13 |
32 | 30, 31 | syl6req 2089 | . . . . . . . . . . . 12 |
33 | 32 | pm4.71ri 372 | . . . . . . . . . . 11 |
34 | 33 | anbi1i 431 | . . . . . . . . . 10 |
35 | anass 381 | . . . . . . . . . 10 | |
36 | 34, 35 | bitri 173 | . . . . . . . . 9 |
37 | 24, 28, 36 | 3bitri 195 | . . . . . . . 8 |
38 | 37 | exbii 1496 | . . . . . . 7 |
39 | 5, 38 | bitri 173 | . . . . . 6 |
40 | opeq1 3549 | . . . . . . . . 9 | |
41 | 40 | eqeq2d 2051 | . . . . . . . 8 |
42 | eleq1 2100 | . . . . . . . 8 | |
43 | 41, 42 | anbi12d 442 | . . . . . . 7 |
44 | 43 | ceqsexgv 2673 | . . . . . 6 |
45 | 39, 44 | syl5bb 181 | . . . . 5 |
46 | 18, 45 | syl 14 | . . . 4 |
47 | 46 | pm5.32ri 428 | . . 3 |
48 | 32 | adantr 261 | . . . . 5 |
49 | 48 | pm4.71i 371 | . . . 4 |
50 | 43 | pm5.32ri 428 | . . . 4 |
51 | 49, 50 | bitr2i 174 | . . 3 |
52 | ancom 253 | . . 3 | |
53 | 47, 51, 52 | 3bitri 195 | . 2 |
54 | 4, 1, 15, 17, 53 | en2i 6250 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 csn 3375 cop 3378 cint 3615 class class class wbr 3764 cxp 4343 cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-en 6222 |
This theorem is referenced by: xpsneng 6296 endisj 6298 |
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