Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ineqcomi | Structured version Visualization version GIF version |
Description: Disjointness inference (when 𝐶 = ∅), inference form of ineqcom 35538. (Contributed by Peter Mazsa, 26-Mar-2017.) |
Ref | Expression |
---|---|
ineqcomi.1 | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Ref | Expression |
---|---|
ineqcomi | ⊢ (𝐵 ∩ 𝐴) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineqcomi.1 | . 2 ⊢ (𝐴 ∩ 𝐵) = 𝐶 | |
2 | ineqcom 35538 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶) | |
3 | 1, 2 | mpbi 232 | 1 ⊢ (𝐵 ∩ 𝐴) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∩ cin 3928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1539 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-rab 3146 df-in 3936 |
This theorem is referenced by: inres2 35540 |
Copyright terms: Public domain | W3C validator |