Theorem dfttc4 37496

Description: An alternative expression for the transitive closure of a class, assuming Regularity. A set 𝑥 is contained in the transitive closure of 𝐴 iff we can construct an ∈-chain from 𝑥 to an element of 𝐴. This weak definition is primarily useful for proving elttcirr 37497. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
dfttc4	⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

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

Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem dfttc4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
2	1	dfttc4lem2 37495	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ∧ Tr {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))})
3		ttcmin 37462	. . 3 ⊢ ((𝐴 ⊆ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ∧ Tr {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}) → TC+ 𝐴 ⊆ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))})
4	2, 3	ax-mp 5	. 2 ⊢ TC+ 𝐴 ⊆ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
5		vex 3458	. . . . 5 ⊢ 𝑤 ∈ V
6		equequ2 2061	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
7	6	imbi2d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
8	7	ralbidv 3189	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
9	8	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤))))
10	9	exbidv 1956	. . . . 5 ⊢ (𝑥 = 𝑤 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤))))
11	5, 10	elab 3636	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
12		vex 3458	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	12	inex2 5282	. . . . . . . 8 ⊢ (TC+ 𝐴 ∩ 𝑦) ∈ V
14		ttcid 37458	. . . . . . . . . 10 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ssrin 4141	. . . . . . . . . 10 ⊢ (𝐴 ⊆ TC+ 𝐴 → (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦))
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦)
17		ssn0 4358	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦) ∧ (𝐴 ∩ 𝑦) ≠ ∅) → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
18	16, 17	mpan 703	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
19		zfreg 9617	. . . . . . . 8 ⊢ (((TC+ 𝐴 ∩ 𝑦) ∈ V ∧ (TC+ 𝐴 ∩ 𝑦) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
20	13, 18, 19	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
21		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∩ 𝑦))
22	21	elin2d 4105	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ 𝑦)
23		inass 4127	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦))
24		elinel1 4101	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ∈ TC+ 𝐴)
25		ttctr2 37460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
26	24, 25	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ⊆ TC+ 𝐴)
27		dfss2 4097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
28	26, 27	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
29	28	ineq1d 4119	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ 𝑦))
30	23, 29	eqtr3id 2812	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = (𝑥 ∩ 𝑦))
31	30	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
32	31	biimpa 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (𝑥 ∩ 𝑦) = ∅)
33		ineq1 4113	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∩ 𝑦) = (𝑥 ∩ 𝑦))
34	33	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
35		equequ1 2060	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑤 ↔ 𝑥 = 𝑤))
36	34, 35	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ↔ ((𝑥 ∩ 𝑦) = ∅ → 𝑥 = 𝑤)))
37	36	rspcv 3576	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∩ 𝑦) = ∅ → 𝑥 = 𝑤)))
38	37	com23 87	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → 𝑥 = 𝑤)))
39	22, 32, 38	sylc 66	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → 𝑥 = 𝑤))
40	39	com12 33	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 = 𝑤))
41		eleq1w 2847	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ↔ 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
42	41	biimpcd 252	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 = 𝑤 → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
43	42	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (𝑥 = 𝑤 → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
44	40, 43	sylcom 31	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
45	44	imp 412	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ∧ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
46	45	rexlimdvaa 3168	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → (∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
47	20, 46	mpan9 516	. . . . . 6 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
48	47	elin1d 4104	. . . . 5 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
49	48	exlimiv 1965	. . . 4 ⊢ (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
50	11, 49	sylbi 220	. . 3 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} → 𝑤 ∈ TC+ 𝐴)
51	50	ssriv 3917	. 2 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ⊆ TC+ 𝐴
52	4, 51	eqssi 3929	1 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  Tr wtr 5215  TC+ cttc 37452

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775  ax-reg 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-ttc 37453

This theorem is used by:  elttcirr  37497