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Theorem sssneq 13368
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} → ∀𝑦𝐴𝑧𝐴 𝑦 = 𝑧)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧

Proof of Theorem sssneq
StepHypRef Expression
1 simpl 108 . . . . 5 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝐴 ⊆ {𝐵})
2 simprl 521 . . . . 5 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
31, 2sseldd 3102 . . . 4 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 ∈ {𝐵})
4 elsni 3549 . . . 4 (𝑦 ∈ {𝐵} → 𝑦 = 𝐵)
53, 4syl 14 . . 3 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 = 𝐵)
6 simprr 522 . . . . 5 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
71, 6sseldd 3102 . . . 4 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑧 ∈ {𝐵})
8 elsni 3549 . . . 4 (𝑧 ∈ {𝐵} → 𝑧 = 𝐵)
97, 8syl 14 . . 3 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑧 = 𝐵)
105, 9eqtr4d 2176 . 2 ((𝐴 ⊆ {𝐵} ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 = 𝑧)
1110ralrimivva 2517 1 (𝐴 ⊆ {𝐵} → ∀𝑦𝐴𝑧𝐴 𝑦 = 𝑧)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  wral 2417  wss 3075  {csn 3531
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3081  df-ss 3088  df-sn 3537
This theorem is referenced by: (None)
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