Theorem sssneq 13716

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 108	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐴 ⊆ {𝐵})
2		simprl 521	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
3	1, 2	sseldd 3138	. . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ {𝐵})
4		elsni 3588	. . . 4 ⊢ (𝑦 ∈ {𝐵} → 𝑦 = 𝐵)
5	3, 4	syl 14	. . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝐵)
6		simprr 522	. . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
7	1, 6	sseldd 3138	. . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ {𝐵})
8		elsni 3588	. . . 4 ⊢ (𝑧 ∈ {𝐵} → 𝑧 = 𝐵)
9	7, 8	syl 14	. . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 = 𝐵)
10	5, 9	eqtr4d 2200	. 2 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝑧)
11	10	ralrimivva 2546	1 ⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342   ∈ wcel 2135  ∀wral 2442   ⊆ wss 3111  {csn 3570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2723  df-in 3117  df-ss 3124  df-sn 3576

This theorem is referenced by: (None)