Theorem nfdisj1 5089

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 5076	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3396	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2357	. 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 2147  ∃*wrmo 3368  Disj wdisj 5075

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 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2567  df-rmo 3369  df-disj 5076

This theorem is used by:  disjabrex  33274  disjabrexf  33275  hasheuni  34828  ldgenpisyslem1  34907  measvunilem  34956  measvunilem0  34957  measvuni  34958  measinblem  34964  voliune  34973  volfiniune  34974  volmeas  34975  dstrvprob  35215  ismeannd  47750