Theorem n0nsnel 33111

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 4287	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅)
2		eqsn 4790	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
3	1, 2	syl 18	. . . . 5 ⊢ (𝐶 ∈ 𝐵 → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
4	3	biimprd 251	. . . 4 ⊢ (𝐶 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝐵 = {𝐴}))
5	4	con3d 153	. . 3 ⊢ (𝐶 ∈ 𝐵 → (¬ 𝐵 = {𝐴} → ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
6		df-ne 2957	. . 3 ⊢ (𝐵 ≠ {𝐴} ↔ ¬ 𝐵 = {𝐴})
7		nne 2960	. . . . . . 7 ⊢ (¬ 𝑥 ≠ 𝐴 ↔ 𝑥 = 𝐴)
8	7	bicomi 227	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ ¬ 𝑥 ≠ 𝐴)
9	8	ralbii 3109	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴)
10		ralnex 3089	. . . . 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 ⊢ (𝐶 ∈ 𝐵 → (𝐵 ≠ {𝐴} → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴))
14	13	imp 412	1 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∅c0 4279  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-dif 3902  df-ss 3916  df-nul 4280  df-sn 4585

This theorem is used by:  krullndrng  34005