Theorem sstrALT2VD 41175

Description: Virtual deduction proof of sstrALT2 41176. (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 3957	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		idn1 40915	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   )
3		simpr 487	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	e1a 40968	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐵 ⊆ 𝐶   )
5		simpl 485	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	e1a 40968	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐵   )
7		idn2 40954	. . . . . . 7 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
8		ssel2 3964	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	e12an 41066	. . . . . 6 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐵   )
10		ssel2 3964	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	e12an 41066	. . . . 5 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐶   )
12	11	in2 40946	. . . 4 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
13	12	gen11 40957	. . 3 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)   )
14		biimpr 222	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	e01 41032	. 2 ⊢ (   (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)   ▶   𝐴 ⊆ 𝐶   )
16	15	in1 40912	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   ∈ wcel 2114   ⊆ wss 3938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-in 3945  df-ss 3954  df-vd1 40911  df-vd2 40919

This theorem is referenced by: (None)