notnotsnex - Intuitionistic Logic Explorer

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Theorem notnotsnex 4166

Description: A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)

**Assertion**
Ref	Expression
notnotsnex

**Proof of Theorem notnotsnex**
Step	Hyp	Ref	Expression
1		snexg 4163	. . . . 5
2	1	con3i 622	. . . 4
3		snexprc 4165	. . . 4
4	2, 3	syl 14	. . 3
5	4	con3i 622	. 2
6		pm2.01 606	. 2
7	5, 6	ax-mp 5	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wcel 2136

cvv 2726

csn 3576

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-nul 4108 ax-pow 4153

This theorem depends on definitions: df-bi 116 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-v 2728 df-dif 3118 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582

This theorem is referenced by: (None)