New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > riinn0 | GIF version |
Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
riinn0 | ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (A ∩ ∩x ∈ X S) = ∩x ∈ X S) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3448 | . 2 ⊢ (A ∩ ∩x ∈ X S) = (∩x ∈ X S ∩ A) | |
2 | r19.2z 3639 | . . . . 5 ⊢ ((X ≠ ∅ ∧ ∀x ∈ X S ⊆ A) → ∃x ∈ X S ⊆ A) | |
3 | 2 | ancoms 439 | . . . 4 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → ∃x ∈ X S ⊆ A) |
4 | iinss 4017 | . . . 4 ⊢ (∃x ∈ X S ⊆ A → ∩x ∈ X S ⊆ A) | |
5 | 3, 4 | syl 15 | . . 3 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → ∩x ∈ X S ⊆ A) |
6 | df-ss 3259 | . . 3 ⊢ (∩x ∈ X S ⊆ A ↔ (∩x ∈ X S ∩ A) = ∩x ∈ X S) | |
7 | 5, 6 | sylib 188 | . 2 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (∩x ∈ X S ∩ A) = ∩x ∈ X S) |
8 | 1, 7 | syl5eq 2397 | 1 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (A ∩ ∩x ∈ X S) = ∩x ∈ X S) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ≠ wne 2516 ∀wral 2614 ∃wrex 2615 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 ∩ciin 3970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-iin 3972 |
This theorem is referenced by: riinrab 4041 |
Copyright terms: Public domain | W3C validator |