Theorem nfdisj1 5128

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 5115	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3408	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2317	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1856	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ∀wal 1540  Ⅎwnf 1786   ∈ wcel 2107  ∃*wrmo 3376  Disj wdisj 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1783  df-nf 1787  df-mo 2535  df-rmo 3377  df-disj 5115

This theorem is referenced by:  disjabrex  31813  disjabrexf  31814  hasheuni  33083  ldgenpisyslem1  33161  measvunilem  33210  measvunilem0  33211  measvuni  33212  measinblem  33218  voliune  33227  volfiniune  33228  volmeas  33229  dstrvprob  33470  ismeannd  45183