Theorem nfdisj1 5092

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
nfdisj1	⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfdisj1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 5079	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3398	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2358	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1886	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2146  ∃*wrmo 3370  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2569  df-rmo 3371  df-disj 5079

This theorem is used by:  disjabrex  33000  disjabrexf  33001  hasheuni  34541  ldgenpisyslem1  34620  measvunilem  34669  measvunilem0  34670  measvuni  34671  measinblem  34677  voliune  34686  volfiniune  34687  volmeas  34688  dstrvprob  34929  ismeannd  47241