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Mirrors > Home > MPE Home > Th. List > rint0 | Structured version Visualization version GIF version |
Description: Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
rint0 | ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteq 4478 | . . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅) | |
2 | 1 | ineq2d 3814 | . 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅)) |
3 | int0 4490 | . . . 4 ⊢ ∩ ∅ = V | |
4 | 3 | ineq2i 3811 | . . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V) |
5 | inv1 3970 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 4, 5 | eqtri 2644 | . 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴 |
7 | 2, 6 | syl6eq 2672 | 1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Vcvv 3200 ∩ cin 3573 ∅c0 3915 ∩ cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-int 4476 |
This theorem is referenced by: incexclem 14568 incexc 14569 mrerintcl 16257 ismred2 16263 txtube 21443 bj-mooreset 33056 bj-ismoored0 33061 bj-ismooredr2 33065 |
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