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Theorem dtruex 4310
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3969 can also be summarized as "at least two sets exist", the difference is that dtruarb 3969 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 2577 . . . . 5 𝑦 ∈ V
2 snexgOLD 3962 . . . . 5 (𝑦 ∈ V → {𝑦} ∈ V)
31, 2ax-mp 7 . . . 4 {𝑦} ∈ V
4 isset 2578 . . . 4 ({𝑦} ∈ V ↔ ∃𝑥 𝑥 = {𝑦})
53, 4mpbi 137 . . 3 𝑥 𝑥 = {𝑦}
6 elirr 4293 . . . . . . . 8 ¬ 𝑦𝑦
7 vsnid 3430 . . . . . . . . 9 𝑦 ∈ {𝑦}
8 eleq2 2117 . . . . . . . . 9 (𝑦 = {𝑦} → (𝑦𝑦𝑦 ∈ {𝑦}))
97, 8mpbiri 161 . . . . . . . 8 (𝑦 = {𝑦} → 𝑦𝑦)
106, 9mto 598 . . . . . . 7 ¬ 𝑦 = {𝑦}
11 eqtr 2073 . . . . . . 7 ((𝑦 = 𝑥𝑥 = {𝑦}) → 𝑦 = {𝑦})
1210, 11mto 598 . . . . . 6 ¬ (𝑦 = 𝑥𝑥 = {𝑦})
13 ancom 257 . . . . . 6 ((𝑦 = 𝑥𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥))
1412, 13mtbi 605 . . . . 5 ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)
1514imnani 635 . . . 4 (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥)
1615eximi 1507 . . 3 (∃𝑥 𝑥 = {𝑦} → ∃𝑥 ¬ 𝑦 = 𝑥)
175, 16ax-mp 7 . 2 𝑥 ¬ 𝑦 = 𝑥
18 eqcom 2058 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1918notbii 604 . . 3 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦)
2019exbii 1512 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦)
2117, 20mpbi 137 1 𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408
This theorem is referenced by:  dtru  4311  eunex  4312  brprcneu  5198
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