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Theorem entr 6671

Description: Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)

**Assertion**
Ref	Expression
entr	$/\$

**Proof of Theorem entr**
Step	Hyp	Ref	Expression
1		ener 6666	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6437	. 2 $/\$
4	3	mptru 1340	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wtru 1332

cvv 2681 class class class wbr 3924

wer 6419

cen 6625

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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-er 6422 df-en 6628

This theorem is referenced by: entri 6673 en2sn 6700 xpsnen2g 6716 enen1 6727 enen2 6728 ssenen 6738 phplem4 6742 snnen2og 6746 php5dom 6750 phplem4on 6754 dif1en 6766 dif1enen 6767 fisbth 6770 diffisn 6780 unsnfidcex 6801 unsnfidcel 6802 f1finf1o 6828 en1eqsn 6829 endjusym 6974 carden2bex 7038 pm54.43 7039 pr2ne 7041 djuen 7060 djuenun 7061 djuassen 7066 frecfzen2 10193 uzennn 10202 hashunlem 10543 hashxp 10565 1nprm 11784 hashdvds 11886 unennn 11899 ennnfonelemen 11923 ennnfonelemim 11926 exmidunben 11928 ctinfom 11930 ctinf 11932 pwf1oexmid 13183