![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfdisj2 | Structured version Visualization version GIF version |
Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
dfdisj2 | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disj 4653 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | df-rmo 2949 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
3 | 2 | albii 1787 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | bitri 264 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∀wal 1521 ∈ wcel 2030 ∃*wmo 2499 ∃*wrmo 2944 Disj wdisj 4652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 |
This theorem depends on definitions: df-bi 197 df-rmo 2949 df-disj 4653 |
This theorem is referenced by: disjss1 4658 nfdisj 4664 invdisj 4670 sndisj 4676 disjxsn 4678 disjss3 4684 vitalilem3 23424 |
Copyright terms: Public domain | W3C validator |