Theorem dtruex 4302

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3962 can also be summarized as "at least two sets exist", the difference is that dtruarb 3962 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 2604	. . . . 5 ⊢ 𝑦 ∈ V
2	1	snex 3957	. . . 4 ⊢ {𝑦} ∈ V
3	2	isseti 2607	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
4		elirrv 4291	. . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦
5		vsnid 3426	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
6		eleq2 2142	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
7	5, 6	mpbiri 166	. . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
8	4, 7	mto 620	. . . . . 6 ⊢ ¬ 𝑦 = {𝑦}
9		eqtr 2098	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
10	8, 9	mto 620	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
11		ancom 262	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
12	10, 11	mtbi 627	. . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
13	12	imnani 657	. . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
14	3, 13	eximii 1533	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
15		equcom 1633	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
16	15	notbii 626	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
17	16	exbii 1536	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	mpbi 143	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by:  dtru  4303  eunex  4304  brprcneu  5191