Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xpen | Unicode version |
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) |
Ref | Expression |
---|---|
xpen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6694 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | bren 6694 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 481 | . . 3 |
7 | relen 6691 | . . . . . . 7 | |
8 | 7 | brrelex1i 4631 | . . . . . 6 |
9 | 7 | brrelex1i 4631 | . . . . . 6 |
10 | xpexg 4702 | . . . . . 6 | |
11 | 8, 9, 10 | syl2an 287 | . . . . 5 |
12 | 11 | ad2antrr 480 | . . . 4 |
13 | simplr 520 | . . . . . 6 | |
14 | f1ofn 5417 | . . . . . . . 8 | |
15 | dffn5im 5516 | . . . . . . . 8 | |
16 | 14, 15 | syl 14 | . . . . . . 7 |
17 | f1oeq1 5405 | . . . . . . 7 | |
18 | 13, 16, 17 | 3syl 17 | . . . . . 6 |
19 | 13, 18 | mpbid 146 | . . . . 5 |
20 | simpr 109 | . . . . . 6 | |
21 | f1ofn 5417 | . . . . . . . 8 | |
22 | dffn5im 5516 | . . . . . . . 8 | |
23 | 21, 22 | syl 14 | . . . . . . 7 |
24 | f1oeq1 5405 | . . . . . . 7 | |
25 | 20, 23, 24 | 3syl 17 | . . . . . 6 |
26 | 20, 25 | mpbid 146 | . . . . 5 |
27 | 19, 26 | xpf1o 6791 | . . . 4 |
28 | f1oeng 6704 | . . . 4 | |
29 | 12, 27, 28 | syl2anc 409 | . . 3 |
30 | 6, 29 | exlimddv 1878 | . 2 |
31 | 3, 30 | exlimddv 1878 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cvv 2712 cop 3564 class class class wbr 3967 cmpt 4027 cxp 4586 wfn 5167 wf1o 5171 cfv 5172 cmpo 5828 cen 6685 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-en 6688 |
This theorem is referenced by: xpdjuen 7155 xpnnen 12193 xpomen 12194 qnnen 12230 |
Copyright terms: Public domain | W3C validator |