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Theorem snssiALTVD 41520
Description: Virtual deduction proof of snssiALT 41521. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssiALTVD (𝐴𝐵 → {𝐴} ⊆ 𝐵)

Proof of Theorem snssiALTVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3904 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 idn1 41267 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn2 41306 . . . . . . 7 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4 velsn 4544 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
53, 4e2bi 41325 . . . . . 6 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6 eleq1a 2888 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
72, 5, 6e12 41417 . . . . 5 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥𝐵   )
87in2 41298 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥𝐵)   )
98gen11 41309 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)   )
10 biimpr 223 . . 3 (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) → {𝐴} ⊆ 𝐵))
111, 9, 10e01 41384 . 2 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
1211in1 41264 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2112  wss 3884  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529  df-vd1 41263  df-vd2 41271
This theorem is referenced by: (None)
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