0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5809

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 4175	. . 3 $/\$
3		nfopab1 3992	. . . . . 6
4	3	nfel2 2292	. . . . 5
5	4	nfn 1636	. . . 4
6		nfopab2 3993	. . . . . . 7
7	6	nfel2 2292	. . . . . 6
8	7	nfn 1636	. . . . 5
9		vex 2684	. . . . . . . 8
10		vex 2684	. . . . . . . 8
11	9, 10	opnzi 4152	. . . . . . 7
12		nesym 2351	. . . . . . . 8
13		pm2.21 606	. . . . . . . 8
14	12, 13	sylbi 120	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 274	. . . . 5 $/\$
17	8, 16	exlimi 1573	. . . 4 $/\$
18	5, 17	exlimi 1573	. . 3 $/\$
19	2, 18	sylbi 120	. 2
20	1, 19	pm2.65i 628	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wceq 1331

wex 1468

wcel 1480

wne 2306

c0 3358

cop 3525

copab 3983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985

This theorem is referenced by: (None)