0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5550

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 3995	. . 3 $/\$
3		nfopab1 3826	. . . . . 6
4	3	nfel2 2190	. . . . 5
5	4	nfn 1548	. . . 4
6		nfopab2 3827	. . . . . . 7
7	6	nfel2 2190	. . . . . 6
8	7	nfn 1548	. . . . 5
9		vex 2560	. . . . . . . 8
10		vex 2560	. . . . . . . 8
11	9, 10	opnzi 3972	. . . . . . 7
12		nesym 2250	. . . . . . . 8
13		pm2.21 547	. . . . . . . 8
14	12, 13	sylbi 114	. . . . . . 7
15	11, 14	ax-mp 7	. . . . . 6
16	15	adantr 261	. . . . 5 $/\$
17	8, 16	exlimi 1485	. . . 4 $/\$
18	5, 17	exlimi 1485	. . 3 $/\$
19	2, 18	sylbi 114	. 2
20	1, 19	pm2.65i 568	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wceq 1243

wex 1381

wcel 1393

wne 2204

c0 3224

cop 3378

copab 3817

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819

This theorem is referenced by: (None)