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Mirrors > Home > NFE Home > Th. List > rintn0 | GIF version |
Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) |
Ref | Expression |
---|---|
rintn0 | ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → (A ∩ ∩X) = ∩X) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3448 | . 2 ⊢ (A ∩ ∩X) = (∩X ∩ A) | |
2 | intssuni2 3951 | . . . 4 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → ∩X ⊆ ∪℘A) | |
3 | ssid 3290 | . . . . 5 ⊢ ℘A ⊆ ℘A | |
4 | sspwuni 4051 | . . . . 5 ⊢ (℘A ⊆ ℘A ↔ ∪℘A ⊆ A) | |
5 | 3, 4 | mpbi 199 | . . . 4 ⊢ ∪℘A ⊆ A |
6 | 2, 5 | syl6ss 3284 | . . 3 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → ∩X ⊆ A) |
7 | df-ss 3259 | . . 3 ⊢ (∩X ⊆ A ↔ (∩X ∩ A) = ∩X) | |
8 | 6, 7 | sylib 188 | . 2 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → (∩X ∩ A) = ∩X) |
9 | 1, 8 | syl5eq 2397 | 1 ⊢ ((X ⊆ ℘A ∧ X ≠ ∅) → (A ∩ ∩X) = ∩X) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ≠ wne 2516 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 ℘cpw 3722 ∪cuni 3891 ∩cint 3926 |
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-pw 3724 df-uni 3892 df-int 3927 |
This theorem is referenced by: (None) |
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