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Mirrors > Home > ILE Home > Th. List > dfdisj2 | 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 3978 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | df-rmo 2463 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
3 | 2 | albii 1470 | . 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | bitri 184 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃*wmo 2027 ∈ wcel 2148 ∃*wrmo 2458 Disj wdisj 3977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 |
This theorem depends on definitions: df-bi 117 df-rmo 2463 df-disj 3978 |
This theorem is referenced by: disjss1 3983 nfdisjv 3989 invdisj 3994 sndisj 3996 disjxsn 3998 |
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