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Mirrors > Home > MPE Home > Th. List > elin3 | Structured version Visualization version GIF version |
Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
elin3.x | ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷) |
Ref | Expression |
---|---|
elin3 | ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3796 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | 1 | anbi1i 731 | . 2 ⊢ ((𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷)) |
3 | elin3.x | . . 3 ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷) | |
4 | 3 | elin2 3801 | . 2 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷)) |
5 | df-3an 1039 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷)) | |
6 | 2, 4, 5 | 3bitr4i 292 | 1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 |
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-3an 1039 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-v 3202 df-in 3581 |
This theorem is referenced by: (None) |
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