![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mo0 | Structured version Visualization version GIF version |
Description: "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mo0 | ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsn 48581 | . . . 4 ⊢ {V} = ∅ | |
2 | 1 | eqcomi 2742 | . . 3 ⊢ ∅ = {V} |
3 | eqeq1 2737 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V})) | |
4 | 2, 3 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 = {V}) |
5 | mosn 48582 | . 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ∃*wmo 2534 Vcvv 3477 ∅c0 4339 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-v 3479 df-sbc 3792 df-dif 3966 df-nul 4340 df-sn 4631 |
This theorem is referenced by: mosssn 48584 mo0sn 48585 f1omo 48612 |
Copyright terms: Public domain | W3C validator |