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Theorem entr 6779

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 6774	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6545	. 2 $/\$
4	3	mptru 1362	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wtru 1354

cvv 2737 class class class wbr 4001

wer 6527

cen 6733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-br 4002 df-opab 4063 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-er 6530 df-en 6736

This theorem is referenced by: entri 6781 en2sn 6808 xpsnen2g 6824 enen1 6835 enen2 6836 ssenen 6846 phplem4 6850 snnen2og 6854 php5dom 6858 phplem4on 6862 dif1en 6874 dif1enen 6875 fisbth 6878 diffisn 6888 unsnfidcex 6914 unsnfidcel 6915 f1finf1o 6941 en1eqsn 6942 endjusym 7090 carden2bex 7183 pm54.43 7184 pr2ne 7186 djuen 7205 djuenun 7206 djuassen 7211 frecfzen2 10420 uzennn 10429 hashunlem 10775 hashxp 10797 1nprm 12104 hashdvds 12211 unennn 12388 ennnfonelemen 12412 ennnfonelemim 12415 exmidunben 12417 ctinfom 12419 ctinf 12421 pwf1oexmid 14553