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Theorem snssiALTVD 39559
Description: Virtual deduction proof of snssiALT 39560. (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 3730 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 idn1 39290 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn2 39338 . . . . . . 7 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4 velsn 4335 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
53, 4e2bi 39357 . . . . . 6 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6 eleq1a 2832 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
72, 5, 6e12 39451 . . . . 5 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥𝐵   )
87in2 39330 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥𝐵)   )
98gen11 39341 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)   )
10 biimpr 210 . . 3 (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) → {𝐴} ⊆ 𝐵))
111, 9, 10e01 39416 . 2 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
1211in1 39287 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628   = wceq 1630  wcel 2137  wss 3713  {csn 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-in 3720  df-ss 3727  df-sn 4320  df-vd1 39286  df-vd2 39294
This theorem is referenced by: (None)
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