Theorem sstrALT2 45576

Description: Virtual deduction proof of sstr 3946, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 45575 using the command file translate_without_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sstrALT2	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Proof of Theorem sstrALT2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3923	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		id 23	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶))
3		simpr 490	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
5		simpl 488	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	syl 18	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
7		idd 25	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
8		ssel2 3933	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	syl6an 697	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
10		ssel2 3933	. . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	syl6an 697	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
12	11	idiALT 45220	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
13	12	alrimiv 1960	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
14		biimpr 223	. 2 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	mpsyl 69	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

This theorem is used by: (None)