notnotsnex - Intuitionistic Logic Explorer

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Theorem notnotsnex 4220

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 4217	. . . . 5
2	1	con3i 633	. . . 4
3		snexprc 4219	. . . 4
4	2, 3	syl 14	. . 3
5	4	con3i 633	. 2
6		pm2.01 617	. 2
7	5, 6	ax-mp 5	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wcel 2167

cvv 2763

csn 3622

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-nul 4159 ax-pow 4207

This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-v 2765 df-dif 3159 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628

This theorem is referenced by: (None)