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Theorem mosn 48661
Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mosn (𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosn
StepHypRef Expression
1 rmosn 4724 . . 3 ∃*𝑥 ∈ {𝐵}⊤
2 rmotru 48652 . . 3 (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
31, 2mpbir 231 . 2 ∃*𝑥 𝑥 ∈ {𝐵}
4 eleq2 2828 . . 3 (𝐴 = {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
54mobidv 2547 . 2 (𝐴 = {𝐵} → (∃*𝑥 𝑥𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵}))
63, 5mpbiri 258 1 (𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wtru 1538  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377  {csn 4631
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-v 3480  df-sbc 3792  df-dif 3966  df-nul 4340  df-sn 4632
This theorem is referenced by:  mo0  48662  mosssn  48663  mo0sn  48664
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