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Mirrors > Home > MPE Home > Th. List > Mathboxes > sstrALT2VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of sstrALT2 40588. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sstrALT2VD | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3840 | . . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) | |
2 | idn1 40327 | . . . . . . 7 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ) | |
3 | simpr 477 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | e1a 40380 | . . . . . 6 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ 𝐵 ⊆ 𝐶 ) |
5 | simpl 475 | . . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐵) | |
6 | 2, 5 | e1a 40380 | . . . . . . 7 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ 𝐴 ⊆ 𝐵 ) |
7 | idn2 40366 | . . . . . . 7 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 ) | |
8 | ssel2 3847 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) | |
9 | 6, 7, 8 | e12an 40478 | . . . . . 6 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐵 ) |
10 | ssel2 3847 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) | |
11 | 4, 9, 10 | e12an 40478 | . . . . 5 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐶 ) |
12 | 11 | in2 40358 | . . . 4 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ) |
13 | 12 | gen11 40369 | . . 3 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ) |
14 | biimpr 212 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)) | |
15 | 1, 13, 14 | e01 40444 | . 2 ⊢ ( (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) ▶ 𝐴 ⊆ 𝐶 ) |
16 | 15 | in1 40324 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1505 ∈ wcel 2050 ⊆ wss 3823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-in 3830 df-ss 3837 df-vd1 40323 df-vd2 40331 |
This theorem is referenced by: (None) |
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