Theorem nfdisj 5128

Description: Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfdisjw 5127 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 5117	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1801	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfcvf 2930	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
4		nfdisj.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
5	4	a1i 11	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝐴)
6	3, 5	nfeld 2915	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝐴)
7		nfdisj.2	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
8	7	nfcri 2895	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
9	8	a1i 11	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 ∈ 𝐵)
10	6, 9	nfand 1895	. . . . . 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 2556	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
13	12	mptru 1544	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
14	13	nfal 2322	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
15	1, 14	nfxfr 1850	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1535  ⊤wtru 1538  Ⅎwnf 1780   ∈ wcel 2106  ∃*wmo 2536  Ⅎwnfc 2888  Disj wdisj 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rmo 3378  df-disj 5116

This theorem is referenced by: (None)