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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 4718 | . . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤ | |
| 2 | rmotru 48728 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤) | |
| 3 | 1, 2 | mpbir 231 | . 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵} | 
| 4 | eleq2 2829 | . . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵})) | |
| 5 | 4 | mobidv 2548 | . 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵})) | 
| 6 | 3, 5 | mpbiri 258 | 1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ∃*wmo 2537 ∃*wrmo 3378 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-v 3481 df-sbc 3788 df-dif 3953 df-nul 4333 df-sn 4626 | 
| This theorem is referenced by: mo0 48738 mosssn 48739 mo0sn 48740 oppcmndclem 48920 termcbasmo 49154 setcsnterm 49161 idfudiag1 49183 | 
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