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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6649 |
. . . 4
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2 | 1 | biimpi 119 |
. . 3
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3 | 2 | adantr 274 |
. 2
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4 | bren 6649 |
. . . . 5
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5 | 4 | biimpi 119 |
. . . 4
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6 | 5 | ad2antlr 481 |
. . 3
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7 | relen 6646 |
. . . . . . 7
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8 | 7 | brrelex1i 4590 |
. . . . . 6
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9 | 7 | brrelex1i 4590 |
. . . . . 6
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10 | xpexg 4661 |
. . . . . 6
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11 | 8, 9, 10 | syl2an 287 |
. . . . 5
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12 | 11 | ad2antrr 480 |
. . . 4
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13 | simplr 520 |
. . . . . 6
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14 | f1ofn 5376 |
. . . . . . . 8
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15 | dffn5im 5475 |
. . . . . . . 8
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16 | 14, 15 | syl 14 |
. . . . . . 7
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17 | f1oeq1 5364 |
. . . . . . 7
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18 | 13, 16, 17 | 3syl 17 |
. . . . . 6
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19 | 13, 18 | mpbid 146 |
. . . . 5
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20 | simpr 109 |
. . . . . 6
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21 | f1ofn 5376 |
. . . . . . . 8
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22 | dffn5im 5475 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | f1oeq1 5364 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 20, 23, 24 | 3syl 17 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 20, 25 | mpbid 146 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 19, 26 | xpf1o 6746 |
. . . 4
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28 | f1oeng 6659 |
. . . 4
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29 | 12, 27, 28 | syl2anc 409 |
. . 3
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30 | 6, 29 | exlimddv 1871 |
. 2
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31 | 3, 30 | exlimddv 1871 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-en 6643 |
This theorem is referenced by: xpdjuen 7091 xpnnen 11943 xpomen 11944 qnnen 11980 |
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