Theorem n0nsn2el 46940

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 4364	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
2		eqsn 4854	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = {𝐴} ↔ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
4	3	biimprd 248	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝐵 = {𝐴}))
5	4	con3d 152	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐵 = {𝐴} → ¬ ∀𝑥 ∈ 𝐵 𝑥 = 𝐴))
6		df-ne 2947	. . 3 ⊢ (𝐵 ≠ {𝐴} ↔ ¬ 𝐵 = {𝐴})
7		nne 2950	. . . . . . 7 ⊢ (¬ 𝑥 ≠ 𝐴 ↔ 𝑥 = 𝐴)
8	7	bicomi 224	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ ¬ 𝑥 ≠ 𝐴)
9	8	ralbii 3099	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ≠ 𝐴)
10		ralnex 3078	. . . . 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 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≠ {𝐴} → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴))
14	13	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∅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-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649

This theorem is referenced by: (None)