Theorem nfdifOLD 4110

Description: Obsolete version of nfdif 4109 as of 14-May-2025. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfdif.1	⊢ Ⅎ𝑥𝐴
nfdif.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfdifOLD	⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Proof of Theorem nfdifOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdif2 3940	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
2		nfdif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4	3	nfn 1857	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
5		nfdif.1	. . 3 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfrabw 3459	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∈ wcel 2109  Ⅎwnfc 2884  {crab 3420   ∖ cdif 3928

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421  df-dif 3934

This theorem is referenced by: (None)