Theorem dfdisj2 5069

Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)

Assertion

Ref	Expression
dfdisj2	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem dfdisj2

Step	Hyp	Ref	Expression
1		df-disj 5068	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		df-rmo 3352	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3	2	albii 1821	. 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))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∃*wmo 2538  ∃*wrmo 3351  Disj wdisj 5067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-rmo 3352  df-disj 5068

This theorem is referenced by:  disjss1  5073  nfdisjw  5079  nfdisj  5080  invdisj  5086  sndisj  5092  disjxsn  5094  disjss3  5099  vitalilem3  25579