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Mirrors > Home > MPE Home > Th. List > sndisj | Structured version Visualization version 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 5114 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) | |
2 | moeq 3702 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝑦 | |
3 | simpr 483 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥}) | |
4 | velsn 4643 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
5 | 3, 4 | sylib 217 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥) |
6 | 5 | equcomd 2020 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦) |
7 | 6 | moimi 2537 | . . 3 ⊢ (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) |
8 | 2, 7 | ax-mp 5 | . 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) |
9 | 1, 8 | mpgbir 1799 | 1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2104 ∃*wmo 2530 {csn 4627 Disj wdisj 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-rmo 3374 df-v 3474 df-sn 4628 df-disj 5113 |
This theorem is referenced by: 0disj 5139 fnpreimac 32163 elrspunidl 32820 sibfof 33637 disjsnxp 44058 vonct 45707 |
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