Theorem nfdisj1 5086

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 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3395	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2356	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1885	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:  ∀wal 1567  Ⅎwnf 1815   ∈ wcel 2146  ∃*wrmo 3367  Disj wdisj 5072

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-11 2196  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-mo 2567  df-rmo 3368  df-disj 5073

This theorem is used by:  disjabrex  33230  disjabrexf  33231  hasheuni  34770  ldgenpisyslem1  34849  measvunilem  34898  measvunilem0  34899  measvuni  34900  measinblem  34906  voliune  34915  volfiniune  34916  volmeas  34917  dstrvprob  35157  ismeannd  47616