Theorem dtruex 4556

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4189 can also be summarized as "at least two sets exist", the difference is that dtruarb 4189 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 2740	. . . . 5 ⊢ 𝑦 ∈ V
2	1	snex 4183	. . . 4 ⊢ {𝑦} ∈ V
3	2	isseti 2745	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
4		elirrv 4545	. . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦
5		vsnid 3624	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
6		eleq2 2241	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
7	5, 6	mpbiri 168	. . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
8	4, 7	mto 662	. . . . . 6 ⊢ ¬ 𝑦 = {𝑦}
9		eqtr 2195	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
10	8, 9	mto 662	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
11		ancom 266	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
12	10, 11	mtbi 670	. . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
13	12	imnani 691	. . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
14	3, 13	eximii 1602	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
15		equcom 1706	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
16	15	notbii 668	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
17	16	exbii 1605	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	mpbi 145	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {csn 3592

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-setind 4534

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598

This theorem is referenced by:  dtru  4557  eunex  4558  brprcneu  5505