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Mirrors > Home > MPE Home > Th. List > Mathboxes > mosn | Structured version Visualization version GIF version |
Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mosn | ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmosn 4621 | . . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤ | |
2 | rmotru 45766 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤) | |
3 | 1, 2 | mpbir 234 | . 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵} |
4 | eleq2 2819 | . . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵})) | |
5 | 4 | mobidv 2548 | . 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵})) |
6 | 3, 5 | mpbiri 261 | 1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 ∃*wmo 2537 ∃*wrmo 3054 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-v 3400 df-sbc 3684 df-dif 3856 df-nul 4224 df-sn 4528 |
This theorem is referenced by: mo0 45775 mosssn 45776 mo0sn 45777 |
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