Theorem nfdisj1 5094

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 5081	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3403	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2362	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1880	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ∀wal 1565  Ⅎwnf 1810   ∈ wcel 2149  ∃*wrmo 3375  Disj wdisj 5080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-mo 2573  df-rmo 3376  df-disj 5081

This theorem is referenced by:  disjabrex  32868  disjabrexf  32869  hasheuni  34420  ldgenpisyslem1  34498  measvunilem  34547  measvunilem0  34548  measvuni  34549  measinblem  34555  voliune  34564  volfiniune  34565  volmeas  34566  dstrvprob  34807  ismeannd  47073