Theorem dtruex 4283

Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3942 can also be summarized as "at least two sets exist", the difference is that dtruarb 3942 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 2560	. . . . 5 ⊢ 𝑦 ∈ V
2		snexgOLD 3935	. . . . 5 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	1, 2	ax-mp 7	. . . 4 ⊢ {𝑦} ∈ V
4		isset 2561	. . . 4 ⊢ ({𝑦} ∈ V ↔ ∃𝑥 𝑥 = {𝑦})
5	3, 4	mpbi 133	. . 3 ⊢ ∃𝑥 𝑥 = {𝑦}
6		elirr 4266	. . . . . . . 8 ⊢ ¬ 𝑦 ∈ 𝑦
7		vsnid 3403	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
8		eleq2 2101	. . . . . . . . 9 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦}))
9	7, 8	mpbiri 157	. . . . . . . 8 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦)
10	6, 9	mto 588	. . . . . . 7 ⊢ ¬ 𝑦 = {𝑦}
11		eqtr 2057	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦})
12	10, 11	mto 588	. . . . . 6 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦})
13		ancom 253	. . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
14	12, 13	mtbi 595	. . . . 5 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
15	14	imnani 625	. . . 4 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
16	15	eximi 1491	. . 3 ⊢ (∃𝑥 𝑥 = {𝑦} → ∃𝑥 ¬ 𝑦 = 𝑥)
17	5, 16	ax-mp 7	. 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥
18		eqcom 2042	. . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
19	18	notbii 594	. . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
20	19	exbii 1496	. 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
21	17, 20	mpbi 133	1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by:  dtru  4284  eunex  4285  brprcneu  5171