0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5933

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 4270	. . 3 $/\$
3		nfopab1 4084	. . . . . 6
4	3	nfel2 2342	. . . . 5
5	4	nfn 1668	. . . 4
6		nfopab2 4085	. . . . . . 7
7	6	nfel2 2342	. . . . . 6
8	7	nfn 1668	. . . . 5
9		vex 2752	. . . . . . . 8
10		vex 2752	. . . . . . . 8
11	9, 10	opnzi 4247	. . . . . . 7
12		nesym 2402	. . . . . . . 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 1604	. . . 4 $/\$
18	5, 17	exlimi 1604	. . 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 1363

wex 1502

wcel 2158

wne 2357

c0 3434

cop 3607

copab 4075

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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221

This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077

This theorem is referenced by: (None)