elzdif0 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elzdif0		Structured version Visualization version Unicode version

Theorem elzdif0 30024

Description: Lemma for qqhval2 30026. (Contributed by Thierry Arnoux, 29-Oct-2017.)

**Assertion**
Ref	Expression
elzdif0	$\$ $\/$

**Proof of Theorem elzdif0**
Step	Hyp	Ref	Expression
1		eldifi 3732	. . 3 $\$
2		eldifn 3733	. . 3 $\$
3		elsng 4191	. . . . 5
4	3	notbid 308	. . . 4
5	4	biimpa 501	. . 3 $/\$
6	1, 2, 5	syl2anc 693	. 2 $\$
7		elz 11379	. . . . 5 $/\$ $\/$ $\/$
8	1, 7	sylib 208	. . . 4 $\$ $/\$ $\/$ $\/$
9	8	simprd 479	. . 3 $\$ $\/$ $\/$
10		3orass 1040	. . 3 $\/$ $\/$ $\/$ $\/$
11	9, 10	sylib 208	. 2 $\$ $\/$ $\/$
12		orel1 397	. 2 $\/$ $\/$ $\/$
13	6, 11, 12	sylc 65	1 $\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990 $\$ cdif 3571

csn 4177

cr 9935

cc0 9936

cneg 10267

cn 11020

cz 11377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 df-z 11378

This theorem is referenced by: (None)