Theorem nfdisj1 5124

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 5111	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3411	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2323	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1853	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ∀wal 1538  Ⅎwnf 1783   ∈ wcel 2108  ∃*wrmo 3379  Disj wdisj 5110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784  df-mo 2540  df-rmo 3380  df-disj 5111

This theorem is referenced by:  disjabrex  32595  disjabrexf  32596  hasheuni  34086  ldgenpisyslem1  34164  measvunilem  34213  measvunilem0  34214  measvuni  34215  measinblem  34221  voliune  34230  volfiniune  34231  volmeas  34232  dstrvprob  34474  ismeannd  46482