0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5915

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 4256	. . 3 $/\$
3		nfopab1 4070	. . . . . 6
4	3	nfel2 2332	. . . . 5
5	4	nfn 1658	. . . 4
6		nfopab2 4071	. . . . . . 7
7	6	nfel2 2332	. . . . . 6
8	7	nfn 1658	. . . . 5
9		vex 2740	. . . . . . . 8
10		vex 2740	. . . . . . . 8
11	9, 10	opnzi 4233	. . . . . . 7
12		nesym 2392	. . . . . . . 8
13		pm2.21 617	. . . . . . . 8
14	12, 13	sylbi 121	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 276	. . . . 5 $/\$
17	8, 16	exlimi 1594	. . . 4 $/\$
18	5, 17	exlimi 1594	. . 3 $/\$
19	2, 18	sylbi 121	. 2
20	1, 19	pm2.65i 639	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wceq 1353

wex 1492

wcel 2148

wne 2347

c0 3422

cop 3595

copab 4061

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4063

This theorem is referenced by: (None)