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Mirrors > Home > ILE Home > Th. List > dtruex | GIF version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4110 can also be summarized as "at least two sets exist", the difference is that dtruarb 4110 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.) |
Ref | Expression |
---|---|
dtruex | ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | 1 | snex 4104 | . . . 4 ⊢ {𝑦} ∈ V |
3 | 2 | isseti 2689 | . . 3 ⊢ ∃𝑥 𝑥 = {𝑦} |
4 | elirrv 4458 | . . . . . . 7 ⊢ ¬ 𝑦 ∈ 𝑦 | |
5 | vsnid 3552 | . . . . . . . 8 ⊢ 𝑦 ∈ {𝑦} | |
6 | eleq2 2201 | . . . . . . . 8 ⊢ (𝑦 = {𝑦} → (𝑦 ∈ 𝑦 ↔ 𝑦 ∈ {𝑦})) | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 ⊢ (𝑦 = {𝑦} → 𝑦 ∈ 𝑦) |
8 | 4, 7 | mto 651 | . . . . . 6 ⊢ ¬ 𝑦 = {𝑦} |
9 | eqtr 2155 | . . . . . 6 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) → 𝑦 = {𝑦}) | |
10 | 8, 9 | mto 651 | . . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) |
11 | ancom 264 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑥 = {𝑦}) ↔ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥)) | |
12 | 10, 11 | mtbi 659 | . . . 4 ⊢ ¬ (𝑥 = {𝑦} ∧ 𝑦 = 𝑥) |
13 | 12 | imnani 680 | . . 3 ⊢ (𝑥 = {𝑦} → ¬ 𝑦 = 𝑥) |
14 | 3, 13 | eximii 1581 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
15 | equcom 1682 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
16 | 15 | notbii 657 | . . 3 ⊢ (¬ 𝑦 = 𝑥 ↔ ¬ 𝑥 = 𝑦) |
17 | 16 | exbii 1584 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ∃𝑥 ¬ 𝑥 = 𝑦) |
18 | 14, 17 | mpbi 144 | 1 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {csn 3522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 |
This theorem is referenced by: dtru 4470 eunex 4471 brprcneu 5407 |
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