Theorem n0nsn2el 47790

Description: If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.)

Assertion

Ref	Expression
n0nsn2el	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem n0nsn2el

Step	Hyp	Ref	Expression
1		ne0i 4293	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
2		eqsn 4794	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
3	1, 2	syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
4	3	biimprd 251	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝐵 = {𝐴}))
5	4	con3d 153	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐵 = {𝐴} → ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
6		df-ne 2958	. . 3 ⊢ (𝐵 ≠ {𝐴} ↔ ¬ 𝐵 = {𝐴})
7		nne 2961	. . . . . . 7 ⊢ (¬ 𝑥 ≠ 𝐴 ↔ 𝑥 = 𝐴)
8	7	bicomi 227	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ ¬ 𝑥 ≠ 𝐴)
9	8	ralbii 3110	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴)
10		ralnex 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
11	9, 10	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
12	11	con2bii 360	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴)
13	5, 6, 12	3imtr4g 299	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≠ {𝐴} → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴))
14	13	imp 411	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  ∅c0 4285  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-sn 4589

This theorem is used by: (None)