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Theorem entr 6731

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 6726	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6497	. 2 $/\$
4	3	mptru 1344	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wtru 1336

cvv 2712 class class class wbr 3967

wer 6479

cen 6685

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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-er 6482 df-en 6688

This theorem is referenced by: entri 6733 en2sn 6760 xpsnen2g 6776 enen1 6787 enen2 6788 ssenen 6798 phplem4 6802 snnen2og 6806 php5dom 6810 phplem4on 6814 dif1en 6826 dif1enen 6827 fisbth 6830 diffisn 6840 unsnfidcex 6866 unsnfidcel 6867 f1finf1o 6893 en1eqsn 6894 endjusym 7042 carden2bex 7126 pm54.43 7127 pr2ne 7129 djuen 7148 djuenun 7149 djuassen 7154 frecfzen2 10335 uzennn 10344 hashunlem 10689 hashxp 10711 1nprm 12006 hashdvds 12111 unennn 12196 ennnfonelemen 12220 ennnfonelemim 12223 exmidunben 12225 ctinfom 12227 ctinf 12229 pwf1oexmid 13642