Theorem dtruex 4663

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4287 can also be summarized as "at least two sets exist", the difference is that dtruarb 4287 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 2806	. . . . 5 ⊢ 𝑦 ∈ V
2	1	snex 4281	. . . 4 ⊢ {𝑦} ∈ V
3	2	isseti 2812	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
4		elirrv 4652	. . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦
5		vsnid 3705	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
6		eleq2 2295	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
7	5, 6	mpbiri 168	. . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
8	4, 7	mto 668	. . . . . 6 ⊢ ¬ 𝑦 = {𝑦}
9		eqtr 2249	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
10	8, 9	mto 668	. . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
11		ancom 266	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
12	10, 11	mtbi 677	. . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
13	12	imnani 698	. . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
14	3, 13	eximii 1651	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
15		equcom 1754	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
16	15	notbii 674	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
17	16	exbii 1654	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	mpbi 145	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679

This theorem is referenced by:  dtru  4664  eunex  4665  brprcneu  5641