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Mirrors > Home > ILE Home > Th. List > entr | Unicode version |
Description: Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
Ref | Expression |
---|---|
entr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ener 6681 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | 2 | ertr 6452 |
. 2
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4 | 3 | mptru 1341 |
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-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-v 2691 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-br 3938 df-opab 3998 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-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-er 6437 df-en 6643 |
This theorem is referenced by: entri 6688 en2sn 6715 xpsnen2g 6731 enen1 6742 enen2 6743 ssenen 6753 phplem4 6757 snnen2og 6761 php5dom 6765 phplem4on 6769 dif1en 6781 dif1enen 6782 fisbth 6785 diffisn 6795 unsnfidcex 6816 unsnfidcel 6817 f1finf1o 6843 en1eqsn 6844 endjusym 6989 carden2bex 7062 pm54.43 7063 pr2ne 7065 djuen 7084 djuenun 7085 djuassen 7090 frecfzen2 10231 uzennn 10240 hashunlem 10582 hashxp 10604 1nprm 11831 hashdvds 11933 unennn 11946 ennnfonelemen 11970 ennnfonelemim 11973 exmidunben 11975 ctinfom 11977 ctinf 11979 pwf1oexmid 13367 |
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