Theorem mosn 46046

Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
mosn	⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosn

Step	Hyp	Ref	Expression
1		rmosn 4652	. . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤
2		rmotru 46038	. . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
3	1, 2	mpbir 230	. 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵}
4		eleq2 2827	. . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
5	4	mobidv 2549	. 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵}))
6	3, 5	mpbiri 257	1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1539  ⊤wtru 1540   ∈ wcel 2108  ∃*wmo 2538  ∃*wrmo 3066  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-v 3424  df-sbc 3712  df-dif 3886  df-nul 4254  df-sn 4559

This theorem is referenced by:  mo0  46047  mosssn  46048  mo0sn  46049