Theorem dtruex 4237

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3933 can also be summarized as "at least two sets exist", the difference is that dtruarb 3933 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 2554	. . . . 5 ⊢ y ∈ V
2		snexgOLD 3926	. . . . 5 ⊢ (y ∈ V → {y} ∈ V)
3	1, 2	ax-mp 7	. . . 4 ⊢ {y} ∈ V
4		isset 2555	. . . 4 ⊢ ({y} ∈ V ↔ ∃x x = {y})
5	3, 4	mpbi 133	. . 3 ⊢ ∃x x = {y}
6		elirr 4224	. . . . . . . 8 ⊢ ¬ y ∈ y
7		ssnid 3395	. . . . . . . . 9 ⊢ y ∈ {y}
8		eleq2 2098	. . . . . . . . 9 ⊢ (y = {y} → (y ∈ y ↔ y ∈ {y}))
9	7, 8	mpbiri 157	. . . . . . . 8 ⊢ (y = {y} → y ∈ y)
10	6, 9	mto 587	. . . . . . 7 ⊢ ¬ y = {y}
11		eqtr 2054	. . . . . . 7 ⊢ ((y = x ∧ x = {y}) → y = {y})
12	10, 11	mto 587	. . . . . 6 ⊢ ¬ (y = x ∧ x = {y})
13		ancom 253	. . . . . 6 ⊢ ((y = x ∧ x = {y}) ↔ (x = {y} ∧ y = x))
14	12, 13	mtbi 594	. . . . 5 ⊢ ¬ (x = {y} ∧ y = x)
15	14	imnani 624	. . . 4 ⊢ (x = {y} → ¬ y = x)
16	15	eximi 1488	. . 3 ⊢ (∃x x = {y} → ∃x ¬ y = x)
17	5, 16	ax-mp 7	. 2 ⊢ ∃x ¬ y = x
18		eqcom 2039	. . . 4 ⊢ (y = x ↔ x = y)
19	18	notbii 593	. . 3 ⊢ (¬ y = x ↔ ¬ x = y)
20	19	exbii 1493	. 2 ⊢ (∃x ¬ y = x ↔ ∃x ¬ x = y)
21	17, 20	mpbi 133	1 ⊢ ∃x ¬ x = y

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  {csn 3367

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  dtru  4238  eunex  4239  brprcneu  5114