notnotsnex - Intuitionistic Logic Explorer

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Theorem notnotsnex 4189

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 4186	. . . . 5
2	1	con3i 632	. . . 4
3		snexprc 4188	. . . 4
4	2, 3	syl 14	. . 3
5	4	con3i 632	. 2
6		pm2.01 616	. 2
7	5, 6	ax-mp 5	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wcel 2148

cvv 2739

csn 3594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176

This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2741 df-dif 3133 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600

This theorem is referenced by: (None)