0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 6013

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 4322	. . 3 $/\$
3		nfopab1 4129	. . . . . 6
4	3	nfel2 2363	. . . . 5
5	4	nfn 1682	. . . 4
6		nfopab2 4130	. . . . . . 7
7	6	nfel2 2363	. . . . . 6
8	7	nfn 1682	. . . . 5
9		vex 2779	. . . . . . . 8
10		vex 2779	. . . . . . . 8
11	9, 10	opnzi 4297	. . . . . . 7
12		nesym 2423	. . . . . . . 8
13		pm2.21 618	. . . . . . . 8
14	12, 13	sylbi 121	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 276	. . . . 5 $/\$
17	8, 16	exlimi 1618	. . . 4 $/\$
18	5, 17	exlimi 1618	. . 3 $/\$
19	2, 18	sylbi 121	. 2
20	1, 19	pm2.65i 640	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wceq 1373

wex 1516

wcel 2178

wne 2378

c0 3468

cop 3646

copab 4120

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-v 2778 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-opab 4122

This theorem is referenced by: (None)