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| Description: A restricted at-most-one quantifier over a singleton is always true. (Contributed by AV, 3-Apr-2023.) | 
| Ref | Expression | 
|---|---|
| rmosn | ⊢ ∃*𝑥 ∈ {𝐴}𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idd 24 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) | |
| 2 | nfsbc1v 3807 | . . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | |
| 3 | sbceq1a 3798 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 4 | 2, 3 | rexsngf 4671 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) | 
| 5 | 2, 3 | reusngf 4673 | . . . 4 ⊢ (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) | 
| 6 | 1, 4, 5 | 3imtr4d 294 | . . 3 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑)) | 
| 7 | rmo5 3399 | . . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑)) | |
| 8 | 6, 7 | sylibr 234 | . 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑) | 
| 9 | rmo0 4361 | . . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑 | |
| 10 | snprc 4716 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 11 | rmoeq1 3415 | . . . 4 ⊢ ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑)) | |
| 12 | 10, 11 | sylbi 217 | . . 3 ⊢ (¬ 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑)) | 
| 13 | 9, 12 | mpbiri 258 | . 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑) | 
| 14 | 8, 13 | pm2.61i 182 | 1 ⊢ ∃*𝑥 ∈ {𝐴}𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ∃!wreu 3377 ∃*wrmo 3378 Vcvv 3479 [wsbc 3787 ∅c0 4332 {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: mosn 48737 | 
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