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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 4283 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 3931 | . . 3 ⊢ ∃𝑥 𝑧 ∈ 𝑥 | |
2 | ax-nul 3883 | . . . 4 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦 | |
3 | sp 1401 | . . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | eximii 1493 | . . 3 ⊢ ∃𝑦 ¬ 𝑧 ∈ 𝑦 |
5 | eeanv 1807 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) ↔ (∃𝑥 𝑧 ∈ 𝑥 ∧ ∃𝑦 ¬ 𝑧 ∈ 𝑦)) | |
6 | 1, 4, 5 | mpbir2an 849 | . 2 ⊢ ∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) |
7 | nelneq2 2139 | . . 3 ⊢ ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ¬ 𝑥 = 𝑦) | |
8 | 7 | 2eximi 1492 | . 2 ⊢ (∃𝑥∃𝑦(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦) → ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦) |
9 | 6, 8 | ax-mp 7 | 1 ⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ∀wal 1241 ∃wex 1381 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: (None) |
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