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Mirrors > Home > MPE Home > Th. List > rmosn | Structured version Visualization version GIF version |
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 3764 | . . . . 5 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | |
3 | sbceq1a 3755 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
4 | 2, 3 | rexsngf 4636 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
5 | 2, 3 | reusngf 4638 | . . . 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 3376 | . . 3 ⊢ (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑)) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑) |
9 | rmo0 4324 | . . 3 ⊢ ∃*𝑥 ∈ ∅ 𝜑 | |
10 | snprc 4683 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
11 | rmoeq1 3394 | . . . 4 ⊢ ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑)) | |
12 | 10, 11 | sylbi 216 | . . 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 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ∃!wreu 3354 ∃*wrmo 3355 Vcvv 3448 [wsbc 3744 ∅c0 4287 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-v 3450 df-sbc 3745 df-dif 3918 df-nul 4288 df-sn 4592 |
This theorem is referenced by: mosn 46971 |
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