Theorem dtruex 4474

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4115 can also be summarized as "at least two sets exist", the difference is that dtruarb 4115 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 2689	. . . . 5 ⊢ 𝑦 ∈ V
2	1	snex 4109	. . . 4 ⊢ {𝑦} ∈ V
3	2	isseti 2694	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
4		elirrv 4463	. . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦
5		vsnid 3557	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
6		eleq2 2203	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
7	5, 6	mpbiri 167	. . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
8	4, 7	mto 651	. . . . . 6 ⊢ ¬ 𝑦 = {𝑦}
9		eqtr 2157	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
10	8, 9	mto 651	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
11		ancom 264	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
12	10, 11	mtbi 659	. . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
13	12	imnani 680	. . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
14	3, 13	eximii 1581	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
15		equcom 1682	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
16	15	notbii 657	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
17	16	exbii 1584	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	mpbi 144	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {csn 3527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533

This theorem is referenced by:  dtru  4475  eunex  4476  brprcneu  5414