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Theorem rmosn 4648
Description: A restricted at-most-one quantifier over a singleton is always true. (Contributed by AV, 3-Apr-2023.)
Assertion
Ref Expression
rmosn ∃*𝑥 ∈ {𝐴}𝜑
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmosn
StepHypRef Expression
1 idd 24 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 nfsbc1v 3791 . . . . 5 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 3782 . . . . 5 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
42, 3rexsngf 4603 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
52, 3reusngf 4605 . . . 4 (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
61, 4, 53imtr4d 296 . . 3 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7 rmo5 3434 . . 3 (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
86, 7sylibr 236 . 2 (𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
9 rmo0 4318 . . 3 ∃*𝑥 ∈ ∅ 𝜑
10 snprc 4646 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
11 rmoeq1 3408 . . . 4 ({𝐴} = ∅ → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
1210, 11sylbi 219 . . 3 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
139, 12mpbiri 260 . 2 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
148, 13pm2.61i 184 1 ∃*𝑥 ∈ {𝐴}𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208   = wceq 1533  wcel 2110  wrex 3139  ∃!wreu 3140  ∃*wrmo 3141  Vcvv 3494  [wsbc 3771  c0 4290  {csn 4560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-v 3496  df-sbc 3772  df-dif 3938  df-nul 4291  df-sn 4561
This theorem is referenced by: (None)
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