Theorem dfttc4 36700

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 36701. (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 2735	. . . 4 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
2	1	dfttc4lem2 36699	. . 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 36666	. . 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 3431	. . . . 5 ⊢ 𝑤 ∈ V
6		equequ2 2028	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
7	6	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
8	7	ralbidv 3158	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
9	8	anbi2d 631	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤))))
10	9	exbidv 1923	. . . . 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 3619	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
12		vex 3431	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	12	inex2 5248	. . . . . . . 8 ⊢ (TC+ 𝐴 ∩ 𝑦) ∈ V
14		ttcid 36662	. . . . . . . . . 10 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ssrin 4172	. . . . . . . . . 10 ⊢ (𝐴 ⊆ TC+ 𝐴 → (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦))
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦)
17		ssn0 4334	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦) ∧ (𝐴 ∩ 𝑦) ≠ ∅) → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
18	16, 17	mpan 691	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
19		zfreg 9500	. . . . . . . 8 ⊢ (((TC+ 𝐴 ∩ 𝑦) ∈ V ∧ (TC+ 𝐴 ∩ 𝑦) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
20	13, 18, 19	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
21		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∩ 𝑦))
22	21	elin2d 4136	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ 𝑦)
23		inass 4158	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦))
24		elinel1 4132	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ∈ TC+ 𝐴)
25		ttctr2 36664	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ⊆ TC+ 𝐴)
27		dfss2 3903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
28	26, 27	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
29	28	ineq1d 4150	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ 𝑦))
30	23, 29	eqtr3id 2784	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = (𝑥 ∩ 𝑦))
31	30	eqeq1d 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
32	31	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (𝑥 ∩ 𝑦) = ∅)
33		ineq1 4144	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∩ 𝑦) = (𝑥 ∩ 𝑦))
34	33	eqeq1d 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
35		equequ1 2027	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑤 ↔ 𝑥 = 𝑤))
36	34, 35	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ↔ ((𝑥 ∩ 𝑦) = ∅ → 𝑥 = 𝑤)))
37	36	rspcv 3558	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∩ 𝑦) = ∅ → 𝑥 = 𝑤)))
38	37	com23 86	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → 𝑥 = 𝑤)))
39	22, 32, 38	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → 𝑥 = 𝑤))
40	39	com12 32	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 = 𝑤))
41		eleq1w 2818	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ↔ 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
42	41	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 = 𝑤 → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
43	42	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (𝑥 = 𝑤 → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
44	40, 43	sylcom 30	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
45	44	imp 406	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ∧ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
46	45	rexlimdvaa 3137	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → (∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
47	20, 46	mpan9 506	. . . . . 6 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
48	47	elin1d 4135	. . . . 5 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
49	48	exlimiv 1932	. . . 4 ⊢ (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
50	11, 49	sylbi 217	. . 3 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} → 𝑤 ∈ TC+ 𝐴)
51	50	ssriv 3921	. 2 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ⊆ TC+ 𝐴
52	4, 51	eqssi 3933	1 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2713   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  Vcvv 3427   ∩ cin 3884   ⊆ wss 3885  ∅c0 4263  Tr wtr 5181  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678  ax-reg 9496

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-ttc 36657

This theorem is referenced by:  elttcirr  36701