0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 6076

Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

**Assertion**
Ref	Expression
0neqopab

**Proof of Theorem 0neqopab**
Step	Hyp	Ref	Expression
1		id 19	. 2
2		elopab 4358	. . 3 $/\$
3		nfopab1 4163	. . . . . 6
4	3	nfel2 2388	. . . . 5
5	4	nfn 1706	. . . 4
6		nfopab2 4164	. . . . . . 7
7	6	nfel2 2388	. . . . . 6
8	7	nfn 1706	. . . . 5
9		vex 2806	. . . . . . . 8
10		vex 2806	. . . . . . . 8
11	9, 10	opnzi 4333	. . . . . . 7
12		nesym 2448	. . . . . . . 8
13		pm2.21 622	. . . . . . . 8
14	12, 13	sylbi 121	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 276	. . . . 5 $/\$
17	8, 16	exlimi 1643	. . . 4 $/\$
18	5, 17	exlimi 1643	. . 3 $/\$
19	2, 18	sylbi 121	. 2
20	1, 19	pm2.65i 644	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wceq 1398

wex 1541

wcel 2202

wne 2403

c0 3496

cop 3676

copab 4154

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-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-opab 4156

This theorem is referenced by: (None)