Theorem dfttc4 36828

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 36829. (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 2752	. . . 4 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}
2	1	dfttc4lem2 36827	. . 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 36794	. . 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 3448	. . . . 5 ⊢ 𝑤 ∈ V
6		equequ2 2036	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
7	6	imbi2d 342	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
8	7	ralbidv 3175	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥) ↔ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
9	8	anbi2d 638	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤))))
10	9	exbidv 1931	. . . . 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 3629	. . . 4 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)))
12		vex 3448	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	12	inex2 5264	. . . . . . . 8 ⊢ (TC+ 𝐴 ∩ 𝑦) ∈ V
14		ttcid 36790	. . . . . . . . . 10 ⊢ 𝐴 ⊆ TC+ 𝐴
15		ssrin 4184	. . . . . . . . . 10 ⊢ (𝐴 ⊆ TC+ 𝐴 → (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦))
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦)
17		ssn0 4348	. . . . . . . . 9 ⊢ (((𝐴 ∩ 𝑦) ⊆ (TC+ 𝐴 ∩ 𝑦) ∧ (𝐴 ∩ 𝑦) ≠ ∅) → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
18	16, 17	mpan 698	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → (TC+ 𝐴 ∩ 𝑦) ≠ ∅)
19		zfreg 9530	. . . . . . . 8 ⊢ (((TC+ 𝐴 ∩ 𝑦) ∈ V ∧ (TC+ 𝐴 ∩ 𝑦) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
20	13, 18, 19	sylancr 595	. . . . . . 7 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)
21		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∩ 𝑦))
22	21	elin2d 4148	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → 𝑥 ∈ 𝑦)
23		inass 4170	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦))
24		elinel1 4144	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ∈ TC+ 𝐴)
25		ttctr2 36792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → 𝑥 ⊆ TC+ 𝐴)
27		dfss2 3913	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
28	26, 27	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
29	28	ineq1d 4162	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ TC+ 𝐴) ∩ 𝑦) = (𝑥 ∩ 𝑦))
30	23, 29	eqtr3id 2801	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = (𝑥 ∩ 𝑦))
31	30	eqeq1d 2754	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → ((𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
32	31	biimpa 479	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅) → (𝑥 ∩ 𝑦) = ∅)
33		ineq1 4156	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∩ 𝑦) = (𝑥 ∩ 𝑦))
34	33	eqeq1d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → ((𝑧 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
35		equequ1 2035	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑤 ↔ 𝑥 = 𝑤))
36	34, 35	imbi12d 346	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ↔ ((𝑥 ∩ 𝑦) = ∅ → 𝑥 = 𝑤)))
37	36	rspcv 3568	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ↔ 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
42	41	biimpcd 251	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) → (𝑥 = 𝑤 → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
43	42	adantr 483	. . . . . . . . . 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 409	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) ∧ (𝑥 ∈ (TC+ 𝐴 ∩ 𝑦) ∧ (𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
46	45	rexlimdvaa 3154	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤) → (∃𝑥 ∈ (TC+ 𝐴 ∩ 𝑦)(𝑥 ∩ (TC+ 𝐴 ∩ 𝑦)) = ∅ → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦)))
47	20, 46	mpan9 513	. . . . . 6 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ (TC+ 𝐴 ∩ 𝑦))
48	47	elin1d 4147	. . . . 5 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
49	48	exlimiv 1940	. . . 4 ⊢ (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑤)) → 𝑤 ∈ TC+ 𝐴)
50	11, 49	sylbi 219	. . 3 ⊢ (𝑤 ∈ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} → 𝑤 ∈ TC+ 𝐴)
51	50	ssriv 3931	. 2 ⊢ {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ⊆ TC+ 𝐴
52	4, 51	eqssi 3943	1 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132  {cab 2730   ≠ wne 2947  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  Tr wtr 5197  TC+ cttc 36784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703  ax-reg 9526

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-ttc 36785

This theorem is referenced by:  elttcirr  36829