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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 4079 | . . 3 |
4 | 1, 3 | xpex 4624 | . 2 |
5 | elxp 4526 | . . 3 | |
6 | inteq 3744 | . . . . . . . 8 | |
7 | 6 | inteqd 3746 | . . . . . . 7 |
8 | vex 2663 | . . . . . . . 8 | |
9 | vex 2663 | . . . . . . . 8 | |
10 | 8, 9 | op1stb 4369 | . . . . . . 7 |
11 | 7, 10 | syl6eq 2166 | . . . . . 6 |
12 | 11, 8 | syl6eqel 2208 | . . . . 5 |
13 | 12 | adantr 274 | . . . 4 |
14 | 13 | exlimivv 1852 | . . 3 |
15 | 5, 14 | sylbi 120 | . 2 |
16 | 8, 2 | opex 4121 | . . 3 |
17 | 16 | a1i 9 | . 2 |
18 | eqvisset 2670 | . . . . 5 | |
19 | ancom 264 | . . . . . . . . . . 11 | |
20 | anass 398 | . . . . . . . . . . 11 | |
21 | velsn 3514 | . . . . . . . . . . . 12 | |
22 | 21 | anbi1i 453 | . . . . . . . . . . 11 |
23 | 19, 20, 22 | 3bitr3i 209 | . . . . . . . . . 10 |
24 | 23 | exbii 1569 | . . . . . . . . 9 |
25 | opeq2 3676 | . . . . . . . . . . . 12 | |
26 | 25 | eqeq2d 2129 | . . . . . . . . . . 11 |
27 | 26 | anbi1d 460 | . . . . . . . . . 10 |
28 | 2, 27 | ceqsexv 2699 | . . . . . . . . 9 |
29 | inteq 3744 | . . . . . . . . . . . . . 14 | |
30 | 29 | inteqd 3746 | . . . . . . . . . . . . 13 |
31 | 8, 2 | op1stb 4369 | . . . . . . . . . . . . 13 |
32 | 30, 31 | syl6req 2167 | . . . . . . . . . . . 12 |
33 | 32 | pm4.71ri 389 | . . . . . . . . . . 11 |
34 | 33 | anbi1i 453 | . . . . . . . . . 10 |
35 | anass 398 | . . . . . . . . . 10 | |
36 | 34, 35 | bitri 183 | . . . . . . . . 9 |
37 | 24, 28, 36 | 3bitri 205 | . . . . . . . 8 |
38 | 37 | exbii 1569 | . . . . . . 7 |
39 | 5, 38 | bitri 183 | . . . . . 6 |
40 | opeq1 3675 | . . . . . . . . 9 | |
41 | 40 | eqeq2d 2129 | . . . . . . . 8 |
42 | eleq1 2180 | . . . . . . . 8 | |
43 | 41, 42 | anbi12d 464 | . . . . . . 7 |
44 | 43 | ceqsexgv 2788 | . . . . . 6 |
45 | 39, 44 | syl5bb 191 | . . . . 5 |
46 | 18, 45 | syl 14 | . . . 4 |
47 | 46 | pm5.32ri 450 | . . 3 |
48 | 32 | adantr 274 | . . . . 5 |
49 | 48 | pm4.71i 388 | . . . 4 |
50 | 43 | pm5.32ri 450 | . . . 4 |
51 | 49, 50 | bitr2i 184 | . . 3 |
52 | ancom 264 | . . 3 | |
53 | 47, 51, 52 | 3bitri 205 | . 2 |
54 | 4, 1, 15, 17, 53 | en2i 6632 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cvv 2660 csn 3497 cop 3500 cint 3741 class class class wbr 3899 cxp 4507 cen 6600 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-en 6603 |
This theorem is referenced by: xpsneng 6684 endisj 6686 |
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