Theorem n0nsnel 32444

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.) (Revised by Thierry Arnoux, 28-May-2025.)

Assertion

Ref	Expression
n0nsnel	⊢ ((𝐶 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem n0nsnel

Step	Hyp	Ref	Expression
1		ne0i 4304	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅)
2		eqsn 4793	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ 𝐵 → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
4	3	biimprd 248	. . . 4 ⊢ (𝐶 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝐵 = {𝐴}))
5	4	con3d 152	. . 3 ⊢ (𝐶 ∈ 𝐵 → (¬ 𝐵 = {𝐴} → ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
6		df-ne 2926	. . 3 ⊢ (𝐵 ≠ {𝐴} ↔ ¬ 𝐵 = {𝐴})
7		nne 2929	. . . . . . 7 ⊢ (¬ 𝑥 ≠ 𝐴 ↔ 𝑥 = 𝐴)
8	7	bicomi 224	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ ¬ 𝑥 ≠ 𝐴)
9	8	ralbii 3075	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴)
10		ralnex 3055	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
11	9, 10	bitri 275	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
12	11	con2bii 357	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴)
13	5, 6, 12	3imtr4g 296	. 2 ⊢ (𝐶 ∈ 𝐵 → (𝐵 ≠ {𝐴} → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴))
14	13	imp 406	1 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∅c0 4296  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3449  df-dif 3917  df-ss 3931  df-nul 4297  df-sn 4590

This theorem is referenced by:  krullndrng  33452