Theorem sssneq 15646

Description: Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)

Assertion

Ref	Expression
sssneq	⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧

Proof of Theorem sssneq

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐴 ⊆ {𝐵})
2		simprl 529	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
3	1, 2	sseldd 3184	. . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ {𝐵})
4		elsni 3640	. . . 4 ⊢ (𝑦 ∈ {𝐵} → 𝑦 = 𝐵)
5	3, 4	syl 14	. . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝐵)
6		simprr 531	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
7	1, 6	sseldd 3184	. . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ {𝐵})
8		elsni 3640	. . . 4 ⊢ (𝑧 ∈ {𝐵} → 𝑧 = 𝐵)
9	7, 8	syl 14	. . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 = 𝐵)
10	5, 9	eqtr4d 2232	. 2 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝑧)
11	10	ralrimivva 2579	1 ⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  {csn 3622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-sn 3628

This theorem is referenced by: (None)