Theorem nfdisj1 5081

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 5068	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3394	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2355	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1873	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ∀wal 1558  Ⅎwnf 1803   ∈ wcel 2142  ∃*wrmo 3366  Disj wdisj 5067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-mo 2566  df-rmo 3367  df-disj 5068

This theorem is referenced by:  disjabrex  32782  disjabrexf  32783  hasheuni  34382  ldgenpisyslem1  34460  measvunilem  34509  measvunilem0  34510  measvuni  34511  measinblem  34517  voliune  34526  volfiniune  34527  volmeas  34528  dstrvprob  34769  ismeannd  47041