Theorem dtruex 4221

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3916 can also be summarized as "at least two sets exist", the difference is that dtruarb 3916 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dtruex	⊢ ∃x ¬ x = y

Distinct variable group:   x,y

Proof of Theorem dtruex

Step	Hyp	Ref	Expression
1		vex 2538	. . . . 5 ⊢ y ∈ V
2		snexgOLD 3909	. . . . 5 ⊢ (y ∈ V → {y} ∈ V)
3	1, 2	ax-mp 7	. . . 4 ⊢ {y} ∈ V
4		isset 2539	. . . 4 ⊢ ({y} ∈ V ↔ ∃x x = {y})
5	3, 4	mpbi 133	. . 3 ⊢ ∃x x = {y}
6		elirr 4208	. . . . . . . 8 ⊢ ¬ y ∈ y
7		ssnid 3378	. . . . . . . . 9 ⊢ y ∈ {y}
8		eleq2 2083	. . . . . . . . 9 ⊢ (y = {y} → (y ∈ y ↔ y ∈ {y}))
9	7, 8	mpbiri 157	. . . . . . . 8 ⊢ (y = {y} → y ∈ y)
10	6, 9	mto 575	. . . . . . 7 ⊢ ¬ y = {y}
11		eqtr 2039	. . . . . . 7 ⊢ ((y = x ∧ x = {y}) → y = {y})
12	10, 11	mto 575	. . . . . 6 ⊢ ¬ (y = x ∧ x = {y})
13		ancom 253	. . . . . 6 ⊢ ((y = x ∧ x = {y}) ↔ (x = {y} ∧ y = x))
14	12, 13	mtbi 582	. . . . 5 ⊢ ¬ (x = {y} ∧ y = x)
15	14	imnani 612	. . . 4 ⊢ (x = {y} → ¬ y = x)
16	15	eximi 1473	. . 3 ⊢ (∃x x = {y} → ∃x ¬ y = x)
17	5, 16	ax-mp 7	. 2 ⊢ ∃x ¬ y = x
18		eqcom 2024	. . . 4 ⊢ (y = x ↔ x = y)
19	18	notbii 581	. . 3 ⊢ (¬ y = x ↔ ¬ x = y)
20	19	exbii 1478	. 2 ⊢ (∃x ¬ y = x ↔ ∃x ¬ x = y)
21	17, 20	mpbi 133	1 ⊢ ∃x ¬ x = y

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  {csn 3350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-setind 4204

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356

This theorem is referenced by:  dtru  4222  eunex  4223  brprcneu  5096