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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 5036 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥})) | |
2 | moeq 3701 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝑦 | |
3 | simpr 487 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥}) | |
4 | velsn 4586 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
5 | 3, 4 | sylib 220 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑦 = 𝑥) |
6 | 5 | equcomd 2025 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝑥}) → 𝑥 = 𝑦) |
7 | 6 | moimi 2626 | . . 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 398 ∈ wcel 2113 ∃*wmo 2619 {csn 4570 Disj wdisj 5034 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rmo 3149 df-v 3499 df-sn 4571 df-disj 5035 |
This theorem is referenced by: 0disj 5061 fnpreimac 30419 sibfof 31602 disjsnxp 41338 vonct 42982 |
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