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Mirrors > Home > MPE Home > Th. List > Mathboxes > mosssn | Structured version Visualization version GIF version |
Description: "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.) |
Ref | Expression |
---|---|
mosssn | ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssn 4724 | . 2 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) | |
2 | mo0 45738 | . . 3 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴) | |
3 | mosn 45737 | . . 3 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) | |
4 | 2, 3 | jaoi 856 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥 ∈ 𝐴) |
5 | 1, 4 | sylbi 220 | 1 ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∃*wmo 2539 ⊆ wss 3853 ∅c0 4221 {csn 4526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-v 3402 df-sbc 3686 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4222 df-sn 4527 |
This theorem is referenced by: (None) |
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