Theorem dtruex 4596

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4225 can also be summarized as "at least two sets exist", the difference is that dtruarb 4225 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 2766	. . . . 5 ⊢ 𝑦 ∈ V
2	1	snex 4219	. . . 4 ⊢ {𝑦} ∈ V
3	2	isseti 2771	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
4		elirrv 4585	. . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦
5		vsnid 3655	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
6		eleq2 2260	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
7	5, 6	mpbiri 168	. . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
8	4, 7	mto 663	. . . . . 6 ⊢ ¬ 𝑦 = {𝑦}
9		eqtr 2214	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
10	8, 9	mto 663	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
11		ancom 266	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
12	10, 11	mtbi 671	. . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
13	12	imnani 692	. . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
14	3, 13	eximii 1616	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
15		equcom 1720	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
16	15	notbii 669	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
17	16	exbii 1619	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	mpbi 145	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {csn 3623

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 4152  ax-pow 4208  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629

This theorem is referenced by:  dtru  4597  eunex  4598  brprcneu  5554