| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssf | Structured version Visualization version GIF version | ||
| Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| iinssf.1 | ⊢ Ⅎ𝑥𝐶 |
| Ref | Expression |
|---|---|
| iinssf | ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eliin 4996 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
| 2 | 1 | elv 3485 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
| 3 | ssel 3977 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
| 4 | 3 | reximi 3084 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
| 5 | iinssf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
| 6 | 5 | nfcri 2897 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶 |
| 7 | 6 | r19.36vf 45141 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
| 8 | 4, 7 | syl 17 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
| 9 | 2, 8 | biimtrid 242 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐶)) |
| 10 | 9 | ssrdv 3989 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 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-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-rex 3071 df-v 3482 df-ss 3968 df-iin 4994 |
| This theorem is referenced by: iinssdf 45144 iinss2d 45162 |
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