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Theorem entr 6802

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 6797	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6568	. 2 $/\$
4	3	mptru 1373	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wtru 1365

cvv 2752 class class class wbr 4018

wer 6550

cen 6756

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-er 6553 df-en 6759

This theorem is referenced by: entri 6804 en2sn 6831 xpsnen2g 6847 enen1 6858 enen2 6859 ssenen 6869 phplem4 6873 snnen2og 6877 php5dom 6881 phplem4on 6885 dif1en 6897 dif1enen 6898 fisbth 6901 diffisn 6911 unsnfidcex 6937 unsnfidcel 6938 f1finf1o 6964 en1eqsn 6965 endjusym 7113 carden2bex 7206 pm54.43 7207 pr2ne 7209 djuen 7228 djuenun 7229 djuassen 7234 frecfzen2 10445 uzennn 10454 hashunlem 10802 hashxp 10824 1nprm 12132 hashdvds 12239 unennn 12416 ennnfonelemen 12440 ennnfonelemim 12443 exmidunben 12445 ctinfom 12447 ctinf 12449 pwf1oexmid 15134