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Theorem snsssn 3691
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1 𝐴 ∈ V
Assertion
Ref Expression
snsssn ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Proof of Theorem snsssn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3086 . . 3 ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}))
2 velsn 3544 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 3544 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 238 . . . 4 ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
54albii 1446 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
61, 5bitri 183 . 2 ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
7 sneqr.1 . . 3 𝐴 ∈ V
8 sbceqal 2964 . . 3 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
97, 8ax-mp 5 . 2 (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
106, 9sylbi 120 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071  {csn 3527
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-in 3077  df-ss 3084  df-sn 3533
This theorem is referenced by: (None)
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