Theorem nfdisj 5146

Description: Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfdisjw 5145 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)

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 5135	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1802	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfcvf 2938	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
4		nfdisj.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	4	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝐴)
6	3, 5	nfeld 2920	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝐴)
7		nfdisj.2	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
8	7	nfcri 2900	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
9	8	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 ∈ 𝐵)
10	6, 9	nfand 1896	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	adantl 481	. . . . 5 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
12	2, 11	nfmod2 2561	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
13	12	mptru 1544	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
14	13	nfal 2327	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
15	1, 14	nfxfr 1851	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1535  ⊤wtru 1538  Ⅎwnf 1781   ∈ wcel 2108  ∃*wmo 2541  Ⅎwnfc 2893  Disj wdisj 5133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rmo 3388  df-disj 5134

This theorem is referenced by: (None)