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Theorem rmosn 4685
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 3764 . . . . 5 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 3755 . . . . 5 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
42, 3rexsngf 4636 . . . 4 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
52, 3reusngf 4638 . . . 4 (𝐴 ∈ V → (∃!𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
61, 4, 53imtr4d 294 . . 3 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
7 rmo5 3376 . . 3 (∃*𝑥 ∈ {𝐴}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 → ∃!𝑥 ∈ {𝐴}𝜑))
86, 7sylibr 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))
1210, 11sylbi 216 . . 3 𝐴 ∈ V → (∃*𝑥 ∈ {𝐴}𝜑 ↔ ∃*𝑥 ∈ ∅ 𝜑))
139, 12mpbiri 258 . 2 𝐴 ∈ V → ∃*𝑥 ∈ {𝐴}𝜑)
148, 13pm2.61i 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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