0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5990

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 4304	. . 3 $/\$
3		nfopab1 4113	. . . . . 6
4	3	nfel2 2361	. . . . 5
5	4	nfn 1681	. . . 4
6		nfopab2 4114	. . . . . . 7
7	6	nfel2 2361	. . . . . 6
8	7	nfn 1681	. . . . 5
9		vex 2775	. . . . . . . 8
10		vex 2775	. . . . . . . 8
11	9, 10	opnzi 4279	. . . . . . 7
12		nesym 2421	. . . . . . . 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 1617	. . . 4 $/\$
18	5, 17	exlimi 1617	. . 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 1515

wcel 2176

wne 2376

c0 3460

cop 3636

copab 4104

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-opab 4106

This theorem is referenced by: (None)