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Theorem entr 6750

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 6745	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6516	. 2 $/\$
4	3	mptru 1352	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wtru 1344

cvv 2726 class class class wbr 3982

wer 6498

cen 6704

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-er 6501 df-en 6707

This theorem is referenced by: entri 6752 en2sn 6779 xpsnen2g 6795 enen1 6806 enen2 6807 ssenen 6817 phplem4 6821 snnen2og 6825 php5dom 6829 phplem4on 6833 dif1en 6845 dif1enen 6846 fisbth 6849 diffisn 6859 unsnfidcex 6885 unsnfidcel 6886 f1finf1o 6912 en1eqsn 6913 endjusym 7061 carden2bex 7145 pm54.43 7146 pr2ne 7148 djuen 7167 djuenun 7168 djuassen 7173 frecfzen2 10362 uzennn 10371 hashunlem 10717 hashxp 10739 1nprm 12046 hashdvds 12153 unennn 12330 ennnfonelemen 12354 ennnfonelemim 12357 exmidunben 12359 ctinfom 12361 ctinf 12363 pwf1oexmid 13879