0neqopab - Intuitionistic Logic Explorer

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Theorem 0neqopab 5898

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 4243	. . 3 $/\$
3		nfopab1 4058	. . . . . 6
4	3	nfel2 2325	. . . . 5
5	4	nfn 1651	. . . 4
6		nfopab2 4059	. . . . . . 7
7	6	nfel2 2325	. . . . . 6
8	7	nfn 1651	. . . . 5
9		vex 2733	. . . . . . . 8
10		vex 2733	. . . . . . . 8
11	9, 10	opnzi 4220	. . . . . . 7
12		nesym 2385	. . . . . . . 8
13		pm2.21 612	. . . . . . . 8
14	12, 13	sylbi 120	. . . . . . 7
15	11, 14	ax-mp 5	. . . . . 6
16	15	adantr 274	. . . . 5 $/\$
17	8, 16	exlimi 1587	. . . 4 $/\$
18	5, 17	exlimi 1587	. . 3 $/\$
19	2, 18	sylbi 120	. 2
20	1, 19	pm2.65i 634	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wceq 1348

wex 1485

wcel 2141

wne 2340

c0 3414

cop 3586

copab 4049

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051

This theorem is referenced by: (None)