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Mirrors > Home > MPE Home > Th. List > Mathboxes > sstrALT2 | Structured version Visualization version GIF version |
Description: Virtual deduction proof of sstr 3978, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 41174 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.) |
Ref | Expression |
---|---|
sstrALT2 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3958 | . 2 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
2 | id 22 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶)) | |
3 | simpr 487 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶) |
5 | simpl 485 | . . . . . . 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 3965 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
9 | 6, 7, 8 | syl6an 682 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
10 | ssel2 3965 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) | |
11 | 4, 9, 10 | syl6an 682 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) |
12 | 11 | idiALT 40817 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) |
13 | 12 | alrimiv 1927 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) |
14 | biimpr 222 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∈ wcel 2113 ⊆ wss 3939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-in 3946 df-ss 3955 |
This theorem is referenced by: (None) |
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