Theorem sstrALT2VD 40587

Description: Virtual deduction proof of sstrALT2 40588. (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		dfss2 3840	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		idn1 40327	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   )
3		simpr 477	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	e1a 40380	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐵 ⊆ 𝐶   )
5		simpl 475	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	e1a 40380	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐵   )
7		idn2 40366	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8		ssel2 3847	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	e12an 40478	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐵   )
10		ssel2 3847	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	e12an 40478	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐶   )
12	11	in2 40358	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
13	12	gen11 40369	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
14		biimpr 212	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	e01 40444	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐶   )
16	15	in1 40324	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   ∈ wcel 2050   ⊆ wss 3823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-in 3830  df-ss 3837  df-vd1 40323  df-vd2 40331

This theorem is referenced by: (None)