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Mirrors > Home > NFE Home > Th. List > iinss | GIF version |
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iinss | ⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . 4 ⊢ y ∈ V | |
2 | eliin 3974 | . . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B) |
4 | ssel 3267 | . . . . 5 ⊢ (B ⊆ C → (y ∈ B → y ∈ C)) | |
5 | 4 | reximi 2721 | . . . 4 ⊢ (∃x ∈ A B ⊆ C → ∃x ∈ A (y ∈ B → y ∈ C)) |
6 | r19.36av 2759 | . . . 4 ⊢ (∃x ∈ A (y ∈ B → y ∈ C) → (∀x ∈ A y ∈ B → y ∈ C)) | |
7 | 5, 6 | syl 15 | . . 3 ⊢ (∃x ∈ A B ⊆ C → (∀x ∈ A y ∈ B → y ∈ C)) |
8 | 3, 7 | syl5bi 208 | . 2 ⊢ (∃x ∈ A B ⊆ C → (y ∈ ∩x ∈ A B → y ∈ C)) |
9 | 8 | ssrdv 3278 | 1 ⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 Vcvv 2859 ⊆ wss 3257 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-iin 3972 |
This theorem is referenced by: riinn0 4040 |
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