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Theorem snsssn 3532
 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 2934 . . 3 ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}))
2 velsn 3392 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 velsn 3392 . . . . 5 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 228 . . . 4 ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴𝑥 = 𝐵))
54albii 1359 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
61, 5bitri 173 . 2 ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
7 sneqr.1 . . 3 𝐴 ∈ V
8 sbceqal 2814 . . 3 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
97, 8ax-mp 7 . 2 (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵)
106, 9sylbi 114 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-in 2924  df-ss 2931  df-sn 3381 This theorem is referenced by: (None)
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