Theorem mo0 48534

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

Assertion

Ref	Expression
mo0	⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem mo0

Step	Hyp	Ref	Expression
1		vsn 48532	. . . 4 ⊢ {V} = ∅
2	1	eqcomi 2749	. . 3 ⊢ ∅ = {V}
3		eqeq1 2744	. . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
4	2, 3	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 = {V})
5		mosn 48533	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∃*wmo 2541  Vcvv 3488  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-v 3490  df-sbc 3805  df-dif 3979  df-nul 4353  df-sn 4649

This theorem is referenced by:  mosssn  48535  mo0sn  48536  f1omo  48563