Theorem dfttc4lem2 37239

Description: Lemma for dfttc4 37240. (Contributed by Matthew House, 6-Apr-2026.)

Hypothesis

Ref	Expression
dfttc4lem2.1	⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

Assertion

Ref	Expression
dfttc4lem2	⊢ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐴,𝑦

Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑥, 𝑦, 𝑧)

Proof of Theorem dfttc4lem2

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjsn 4671	. . . . . 6 ⊢ ((𝐴 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢 ∈ 𝐴)
2	1	biimpi 219	. . . . 5 ⊢ ((𝐴 ∩ {𝑢}) = ∅ → ¬ 𝑢 ∈ 𝐴)
3	2	necon2ai 2984	. . . 4 ⊢ (𝑢 ∈ 𝐴 → (𝐴 ∩ {𝑢}) ≠ ∅)
4		elsni 4600	. . . . . 6 ⊢ (𝑧 ∈ {𝑢} → 𝑧 = 𝑢)
5	4	a1d 26	. . . . 5 ⊢ (𝑧 ∈ {𝑢} → ((𝑧 ∩ {𝑢}) = ∅ → 𝑧 = 𝑢))
6	5	rgen 3078	. . . 4 ⊢ ∀𝑧 ∈ {𝑢} ((𝑧 ∩ {𝑢}) = ∅ → 𝑧 = 𝑢)
7		dfttc4lem2.1	. . . . 5 ⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
8		vsnex 5392	. . . . 5 ⊢ {𝑢} ∈ V
9		vex 3454	. . . . 5 ⊢ 𝑢 ∈ V
10	7, 8, 9	dfttc4lem1 37238	. . . 4 ⊢ (((𝐴 ∩ {𝑢}) ≠ ∅ ∧ ∀𝑧 ∈ {𝑢} ((𝑧 ∩ {𝑢}) = ∅ → 𝑧 = 𝑢)) → 𝑢 ∈ 𝐵)
11	3, 6, 10	sylancl 598	. . 3 ⊢ (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵)
12	11	ssriv 3934	. 2 ⊢ 𝐴 ⊆ 𝐵
13		vex 3454	. . . . . . 7 ⊢ 𝑣 ∈ V
14		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
15	14	ineq2d 4165	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝑤))
16	15	neeq1d 3014	. . . . . . . . 9 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ((𝐴 ∩ 𝑦) ≠ ∅ ↔ (𝐴 ∩ 𝑤) ≠ ∅))
17	14	ineq2d 4165	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑤))
18	17	eqeq1d 2762	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑧 ∩ 𝑤) = ∅))
19		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑣)
20	19	eqeq2d 2771	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝑧 = 𝑥 ↔ 𝑧 = 𝑣))
21	18, 20	imbi12d 347	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)))
22	14, 21	raleqbidvv 3327	. . . . . . . . 9 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)))
23	16, 22	anbi12d 644	. . . . . . . 8 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣))))
24	23	cbvexdvaw 2072	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ∃𝑤((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣))))
25	13, 24, 7	elab2 3635	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↔ ∃𝑤((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)))
26		undisj2 4415	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝑤) = ∅ ∧ (𝐴 ∩ {𝑢}) = ∅) ↔ (𝐴 ∩ (𝑤 ∪ {𝑢})) = ∅)
27	26	biimpri 231	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ (𝑤 ∪ {𝑢})) = ∅ → ((𝐴 ∩ 𝑤) = ∅ ∧ (𝐴 ∩ {𝑢}) = ∅))
28	27	simpld 500	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ (𝑤 ∪ {𝑢})) = ∅ → (𝐴 ∩ 𝑤) = ∅)
29	28	necon3i 2987	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝑤) ≠ ∅ → (𝐴 ∩ (𝑤 ∪ {𝑢})) ≠ ∅)
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑣 → ((𝐴 ∩ 𝑤) ≠ ∅ → (𝐴 ∩ (𝑤 ∪ {𝑢})) ≠ ∅))
31		undisj2 4415	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∩ 𝑤) = ∅ ∧ (𝑧 ∩ {𝑢}) = ∅) ↔ (𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅)
32	31	biimpri 231	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → ((𝑧 ∩ 𝑤) = ∅ ∧ (𝑧 ∩ {𝑢}) = ∅))
33	32	simpld 500	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → (𝑧 ∩ 𝑤) = ∅)
34	33	imim1i 64	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣) → ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑣))
35	32	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → (𝑧 ∩ {𝑢}) = ∅)
36		disjsn 4671	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢 ∈ 𝑧)
37	35, 36	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → ¬ 𝑢 ∈ 𝑧)
38		elequ2 2160	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ 𝑣))
39	38	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑣 → (𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑧))
40	39	con3d 153	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑣 → (¬ 𝑢 ∈ 𝑧 → ¬ 𝑢 ∈ 𝑣))
41		pm2.21 124	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑢 ∈ 𝑣 → (𝑢 ∈ 𝑣 → 𝑧 = 𝑢))
42	37, 40, 41	syl56 37	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑣 → ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → (𝑢 ∈ 𝑣 → 𝑧 = 𝑢)))
43	34, 42	syli 40	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣) → ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → (𝑢 ∈ 𝑣 → 𝑧 = 𝑢)))
44	43	com3r 88	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑣 → (((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣) → ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)))
45	44	ralimdv 3176	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑣 → (∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣) → ∀𝑧 ∈ 𝑤 ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)))
46	4	a1d 26	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑢} → ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢))
47	46	rgen 3078	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ {𝑢} ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)
48		ralun 4143	. . . . . . . . . . 11 ⊢ ((∀𝑧 ∈ 𝑤 ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢) ∧ ∀𝑧 ∈ {𝑢} ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)) → ∀𝑧 ∈ (𝑤 ∪ {𝑢})((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢))
49	47, 48	mpan2 704	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑤 ((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢) → ∀𝑧 ∈ (𝑤 ∪ {𝑢})((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢))
50	45, 49	syl6 36	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑣 → (∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣) → ∀𝑧 ∈ (𝑤 ∪ {𝑢})((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)))
51	30, 50	anim12d 621	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑣 → (((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)) → ((𝐴 ∩ (𝑤 ∪ {𝑢})) ≠ ∅ ∧ ∀𝑧 ∈ (𝑤 ∪ {𝑢})((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢))))
52		vex 3454	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
53	52, 8	unex 7744	. . . . . . . . 9 ⊢ (𝑤 ∪ {𝑢}) ∈ V
54	7, 53, 9	dfttc4lem1 37238	. . . . . . . 8 ⊢ (((𝐴 ∩ (𝑤 ∪ {𝑢})) ≠ ∅ ∧ ∀𝑧 ∈ (𝑤 ∪ {𝑢})((𝑧 ∩ (𝑤 ∪ {𝑢})) = ∅ → 𝑧 = 𝑢)) → 𝑢 ∈ 𝐵)
55	51, 54	syl6 36	. . . . . . 7 ⊢ (𝑢 ∈ 𝑣 → (((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)) → 𝑢 ∈ 𝐵))
56	55	exlimdv 1966	. . . . . 6 ⊢ (𝑢 ∈ 𝑣 → (∃𝑤((𝐴 ∩ 𝑤) ≠ ∅ ∧ ∀𝑧 ∈ 𝑤 ((𝑧 ∩ 𝑤) = ∅ → 𝑧 = 𝑣)) → 𝑢 ∈ 𝐵))
57	25, 56	biimtrid 245	. . . . 5 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ 𝐵 → 𝑢 ∈ 𝐵))
58	57	imp 412	. . . 4 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ 𝐵)
59	58	gen2 1829	. . 3 ⊢ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ 𝐵)
60		dftr2 5213	. . 3 ⊢ (Tr 𝐵 ↔ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ 𝐵))
61	59, 60	mpbir 234	. 2 ⊢ Tr 𝐵
62	12, 61	pm3.2i 476	1 ⊢ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  {csn 4583  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-sn 4584  df-pr 4586  df-uni 4867  df-tr 5212

This theorem is used by:  dfttc4  37240