entr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > entr		Unicode version

Theorem entr 6780

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 6775	. . . 4
2	1	a1i 9	. . 3
3	2	ertr 6546	. 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 4002

wer 6528

cen 6734

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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432

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 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-er 6531 df-en 6737

This theorem is referenced by: entri 6782 en2sn 6809 xpsnen2g 6825 enen1 6836 enen2 6837 ssenen 6847 phplem4 6851 snnen2og 6855 php5dom 6859 phplem4on 6863 dif1en 6875 dif1enen 6876 fisbth 6879 diffisn 6889 unsnfidcex 6915 unsnfidcel 6916 f1finf1o 6942 en1eqsn 6943 endjusym 7091 carden2bex 7184 pm54.43 7185 pr2ne 7187 djuen 7206 djuenun 7207 djuassen 7212 frecfzen2 10421 uzennn 10430 hashunlem 10776 hashxp 10798 1nprm 12105 hashdvds 12212 unennn 12389 ennnfonelemen 12413 ennnfonelemim 12416 exmidunben 12418 ctinfom 12420 ctinf 12422 pwf1oexmid 14600