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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 5025 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) | |
2 | moeq 3697 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝑦 | |
3 | simpr 487 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥}) | |
4 | velsn 4576 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
5 | 3, 4 | sylib 220 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥) |
6 | 5 | equcomd 2022 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦) |
7 | 6 | moimi 2623 | . . 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 1796 | 1 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2110 ∃*wmo 2616 {csn 4560 Disj wdisj 5023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rmo 3146 df-v 3496 df-sn 4561 df-disj 5024 |
This theorem is referenced by: 0disj 5050 fnpreimac 30410 sibfof 31593 disjsnxp 41325 vonct 42969 |
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