notnotsnex - Intuitionistic Logic Explorer

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Theorem notnotsnex 4277

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 4274	. . . . 5
2	1	con3i 637	. . . 4
3		snexprc 4276	. . . 4
4	2, 3	syl 14	. . 3
5	4	con3i 637	. 2
6		pm2.01 621	. 2
7	5, 6	ax-mp 5	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wcel 2202

cvv 2802

csn 3669

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-nul 4215 ax-pow 4264

This theorem depends on definitions: df-bi 117 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-v 2804 df-dif 3202 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675

This theorem is referenced by: (None)