Theorem nfdisj 4632

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
nfdisj.1	⊢ Ⅎ𝑦𝐴
nfdisj.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfdisj	⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfdisj

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 4622	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1730	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfcvf 2788	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
4		nfdisj.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	4	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝐴)
6	3, 5	nfeld 2773	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝐴)
7		nfdisj.2	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
8	7	nfcri 2758	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
9	8	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 ∈ 𝐵)
10	6, 9	nfand 1826	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	adantl 482	. . . . 5 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
12	2, 11	nfmod2 2483	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
13	12	trud 1493	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
14	13	nfal 2153	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
15	1, 14	nfxfr 1779	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ¬ wn 3   ∧ wa 384  ∀wal 1481  ⊤wtru 1484  Ⅎwnf 1708   ∈ wcel 1990  ∃*wmo 2471  Ⅎwnfc 2751  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-disj 4621

This theorem is referenced by:  disjxiun  4649  disjxiunOLD  4650