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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 5111 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
| 2 | df-rmo 3380 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
| 3 | 2 | albii 1819 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ∃*wmo 2538 ∃*wrmo 3379 Disj wdisj 5110 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-rmo 3380 df-disj 5111 |
| This theorem is referenced by: disjss1 5116 nfdisjw 5122 nfdisj 5123 invdisj 5129 sndisj 5135 disjxsn 5137 disjss3 5142 vitalilem3 25645 |
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