Theorem nfdifOLD 4080

Description: Obsolete version of nfdif 4079 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 3908	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
2		nfdif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2888	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4	3	nfn 1858	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
5		nfdif.1	. . 3 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfrabw 3434	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
7	1, 6	nfcxfr 2894	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∈ wcel 2113  Ⅎwnfc 2881  {crab 3397   ∖ cdif 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rab 3398  df-dif 3902

This theorem is referenced by: (None)