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Mirrors > Home > MPE Home > Th. List > ssiinf | Structured version Visualization version GIF version |
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ssiinf.1 | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
ssiinf | ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliin 4915 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
2 | 1 | elv 3497 | . . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
3 | 2 | ralbii 3162 | . . 3 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
4 | ssiinf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
5 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
6 | 4, 5 | ralcomf 3354 | . . 3 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) |
7 | 3, 6 | bitri 276 | . 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) |
8 | dfss3 3953 | . 2 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) | |
9 | dfss3 3953 | . . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) | |
10 | 9 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) |
11 | 7, 8, 10 | 3bitr4i 304 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ∩ ciin 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-v 3494 df-in 3940 df-ss 3949 df-iin 4913 |
This theorem is referenced by: ssiin 4970 dmiin 5818 |
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