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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
StepHypRef Expression
1 vex 2560 . . . . 5 𝑦 ∈ V
2 snexgOLD 3935 . . . . 5 (𝑦 ∈ V → {𝑦} ∈ V)
31, 2ax-mp 7 . . . 4 {𝑦} ∈ V
4 isset 2561 . . . 4 ({𝑦} ∈ V ↔ ∃𝑥 𝑥 = {𝑦})
53, 4mpbi 133 . . 3 𝑥 𝑥 = {𝑦}
6 elirr 4266 . . . . . . . 8 ¬ 𝑦𝑦
7 vsnid 3403 . . . . . . . . 9 𝑦 ∈ {𝑦}
8 eleq2 2101 . . . . . . . . 9 (𝑦 = {𝑦} → (𝑦𝑦𝑦 ∈ {𝑦}))
97, 8mpbiri 157 . . . . . . . 8 (𝑦 = {𝑦} → 𝑦𝑦)
106, 9mto 588 . . . . . . 7 ¬ 𝑦 = {𝑦}
11 eqtr 2057 . . . . . . 7 ((𝑦 = 𝑥𝑥 = {𝑦}) → 𝑦 = {𝑦})
1210, 11mto 588 . . . . . 6 ¬ (𝑦 = 𝑥𝑥 = {𝑦})
13 ancom 253 . . . . . 6 ((𝑦 = 𝑥𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
1412, 13mtbi 595 . . . . 5 ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
1514imnani 625 . . . 4 (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
1615eximi 1491 . . 3 (∃𝑥 𝑥 = {𝑦} → ∃𝑥 ¬ 𝑦 = 𝑥)
175, 16ax-mp 7 . 2 𝑥 ¬ 𝑦 = 𝑥
18 eqcom 2042 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1918notbii 594 . . 3 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
2019exbii 1496 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
2117, 20mpbi 133 1 𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff set class 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
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