Theorem sstrALT2VD 45620

Description: Virtual deduction proof of sstrALT2 45621. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sstrALT2VD	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Proof of Theorem sstrALT2VD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3923	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		idn1 45361	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   )
3		simpr 490	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	e1a 45414	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐵 ⊆ 𝐶   )
5		simpl 488	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	e1a 45414	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐵   )
7		idn2 45400	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8		ssel2 3933	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	e12an 45511	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐵   )
10		ssel2 3933	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	e12an 45511	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐶   )
12	11	in2 45392	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
13	12	gen11 45403	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
14		biimpr 223	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	e01 45478	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐶   )
16	15	in1 45358	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2146   ⊆ wss 3906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2840  df-ss 3923  df-vd1 45357  df-vd2 45365

This theorem is used by: (None)