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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 5032 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | df-rmo 3146 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
3 | 2 | albii 1820 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | bitri 277 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1535 ∈ wcel 2114 ∃*wmo 2620 ∃*wrmo 3141 Disj wdisj 5031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-rmo 3146 df-disj 5032 |
This theorem is referenced by: disjss1 5037 nfdisjw 5043 nfdisj 5044 invdisj 5050 sndisj 5057 disjxsn 5059 disjss3 5065 vitalilem3 24211 |
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