Theorem sstrALT2 40727

Description: Virtual deduction proof of sstr 3897, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 40726 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		dfss2 3877	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		id 22	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶))
3		simpr 485	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶)
5		simpl 483	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
6	2, 5	syl 17	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
7		idd 24	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
8		ssel2 3884	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
9	6, 7, 8	syl6an 680	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
10		ssel2 3884	. . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
11	4, 9, 10	syl6an 680	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
12	11	idiALT 40369	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
13	12	alrimiv 1905	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
14		biimpr 221	. 2 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶))
15	1, 13, 14	mpsyl 68	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   ∈ wcel 2081   ⊆ wss 3859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-in 3866  df-ss 3874

This theorem is referenced by: (None)