0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5824

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 4188	. . 3 $/\$
3		nfopab1 4005	. . . . . 6
4	3	nfel2 2295	. . . . 5
5	4	nfn 1637	. . . 4
6		nfopab2 4006	. . . . . . 7
7	6	nfel2 2295	. . . . . 6
8	7	nfn 1637	. . . . 5
9		vex 2692	. . . . . . . 8
10		vex 2692	. . . . . . . 8
11	9, 10	opnzi 4165	. . . . . . 7
12		nesym 2354	. . . . . . . 8
13		pm2.21 607	. . . . . . . 8
14	12, 13	sylbi 120	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 274	. . . . 5 $/\$
17	8, 16	exlimi 1574	. . . 4 $/\$
18	5, 17	exlimi 1574	. . 3 $/\$
19	2, 18	sylbi 120	. 2
20	1, 19	pm2.65i 629	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wceq 1332

wex 1469

wcel 1481

wne 2309

c0 3368

cop 3535

copab 3996

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998

This theorem is referenced by: (None)