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Mirrors > Home > ILE Home > Th. List > dtruarb | GIF version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4469 in which we are given a set 𝑦 and go from there to a set 𝑥 which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.) |
Ref | Expression |
---|---|
dtruarb | ⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 4097 | . . 3 ⊢ ∃𝑥 𝑧 ∈ 𝑥 | |
2 | ax-nul 4049 | . . . 4 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦 | |
3 | sp 1488 | . . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | eximii 1581 | . . 3 ⊢ ∃𝑦 ¬ 𝑧 ∈ 𝑦 |
5 | eeanv 1902 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) ↔ (∃𝑥 𝑧 ∈ 𝑥 ∧ ∃𝑦 ¬ 𝑧 ∈ 𝑦)) | |
6 | 1, 4, 5 | mpbir2an 926 | . 2 ⊢ ∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) |
7 | nelneq2 2239 | . . 3 ⊢ ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ¬ 𝑥 = 𝑦) | |
8 | 7 | 2eximi 1580 | . 2 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦) |
9 | 6, 8 | ax-mp 5 | 1 ⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∀wal 1329 ∃wex 1468 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2119 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2130 df-clel 2133 |
This theorem is referenced by: (None) |
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