Theorem dfdisj2 4108

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 4107	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		df-rmo 2536	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3	2	albii 1523	. 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))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃*wmo 2087   ∈ wcel 2209  ∃*wrmo 2531  Disj wdisj 4106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502

This proof depends on definitions:  df-bi 117  df-rmo 2536  df-disj 4107

This theorem is used by:  disjss1  4112  nfdisjv  4118  invdisj  4123  sndisj  4126  disjxsn  4128