Theorem sstrALT2VD 38549

Description: Virtual deduction proof of sstrALT2 38550. (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 3572	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		idn1 38269	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   )
3		simpr 477	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	e1a 38331	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐵 ⊆ 𝐶   )
5		simpl 473	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	e1a 38331	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐵   )
7		idn2 38317	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8		ssel2 3578	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	e12an 38431	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐵   )
10		ssel2 3578	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	e12an 38431	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐶   )
12	11	in2 38309	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
13	12	gen11 38320	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
14		biimpr 210	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	e01 38395	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐶   )
16	15	in1 38266	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   ∈ wcel 1987   ⊆ wss 3555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3562  df-ss 3569  df-vd1 38265  df-vd2 38273

This theorem is referenced by: (None)