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| 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 4996 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
| 2 | 1 | elv 3485 | . . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | 
| 3 | 2 | ralbii 3093 | . . 3 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | 
| 4 | ssiinf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
| 6 | 4, 5 | ralcomf 3302 | . . 3 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) | 
| 7 | 3, 6 | bitri 275 | . 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) | 
| 8 | dfss3 3972 | . 2 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) | |
| 9 | dfss3 3972 | . . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) | |
| 10 | 9 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵) | 
| 11 | 7, 8, 10 | 3bitr4i 303 | 1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∩ ciin 4992 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-v 3482 df-ss 3968 df-iin 4994 | 
| This theorem is referenced by: ssiin 5055 dmiin 5964 | 
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