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Mirrors > Home > ILE Home > Th. List > sndisj | GIF version |
Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
sndisj | ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 3747 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) | |
2 | moeq 2716 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝑦 | |
3 | simpr 103 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥}) | |
4 | velsn 3392 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
5 | 3, 4 | sylib 127 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥) |
6 | 5 | eqcomd 2045 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦) |
7 | 6 | moimi 1965 | . . 3 ⊢ (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) |
8 | 2, 7 | ax-mp 7 | . 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) |
9 | 1, 8 | mpgbir 1342 | 1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1393 ∃*wmo 1901 {csn 3375 Disj wdisj 3745 |
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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rmo 2314 df-v 2559 df-sn 3381 df-disj 3746 |
This theorem is referenced by: 0disj 3761 |
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