Theorem mosn 47961

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 4728	. . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤
2		rmotru 47953	. . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤)
3	1, 2	mpbir 230	. 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵}
4		eleq2 2818	. . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
5	4	mobidv 2538	. 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 1533  ⊤wtru 1534   ∈ wcel 2098  ∃*wmo 2527  ∃*wrmo 3373  {csn 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-v 3475  df-sbc 3779  df-dif 3952  df-nul 4327  df-sn 4633

This theorem is referenced by:  mo0  47962  mosssn  47963  mo0sn  47964