notnotsnex - Intuitionistic Logic Explorer

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Theorem notnotsnex 4322

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 4319	. . . . 5
2	1	con3i 641	. . . 4
3		snexprc 4321	. . . 4
4	2, 3	syl 14	. . 3
5	4	con3i 641	. 2
6		pm2.01 625	. 2
7	5, 6	ax-mp 5	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wcel 2209

cvv 2821

csn 3708

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-nul 4257 ax-pow 4309

This theorem depends on definitions: df-bi 117 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-v 2823 df-dif 3222 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3714

This theorem is referenced by: (None)