Theorem ttctr 36948

Description: The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttctr	⊢ Tr TC+ 𝐴

Proof of Theorem ttctr

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

Step	Hyp	Ref	Expression
1		rdgfun 8402	. . . . . . . . 9 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
2		eluniima 7248	. . . . . . . . 9 ⊢ (Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) → (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
4		peano2 7885	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
5		elunii 4876	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
6		nnon 7867	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
7		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
8	7	uniex 7739	. . . . . . . . . . . . . . . 16 ⊢ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V
9		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) = rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})
10		unieq 4882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → ∪ 𝑤 = ∪ 𝑦)
11		unieq 4882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → ∪ 𝑤 = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
12	9, 10, 11	rdgsucmpt2 8416	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ On ∧ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
13	6, 8, 12	sylancl 597	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ω → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) = ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))
14	13	eleq2d 2847	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧) ↔ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)))
15	14	biimpar 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧)) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
16	5, 15	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
17		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑧 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧))
18	17	eleq2d 2847	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤) ↔ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)))
19	18	rspcev 3580	. . . . . . . . . . . 12 ⊢ ((suc 𝑧 ∈ ω ∧ 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘suc 𝑧)) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
20	4, 16, 19	syl2an2r 697	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
21		eluniima 7248	. . . . . . . . . . . 12 ⊢ (Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) → (𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤)))
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑤 ∈ ω 𝑢 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑤))
23	20, 22	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ω ∧ (𝑢 ∈ 𝑣 ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
24	23	an12s 661	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑣 ∧ (𝑧 ∈ ω ∧ 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧))) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
25	24	rexlimdvaa 3165	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑣 → (∃𝑧 ∈ ω 𝑣 ∈ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥})‘𝑧) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
26	3, 25	biimtrid 245	. . . . . . 7 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
27	26	reximdv 3178	. . . . . 6 ⊢ (𝑢 ∈ 𝑣 → (∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
28		eliun 4959	. . . . . 6 ⊢ (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑣 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
29		eliun 4959	. . . . . 6 ⊢ (𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) ↔ ∃𝑥 ∈ 𝐴 𝑢 ∈ ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
30	27, 28, 29	3imtr4g 299	. . . . 5 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω) → 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)))
31		df-ttc 36942	. . . . . 6 ⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
32	31	eleq2i 2853	. . . . 5 ⊢ (𝑣 ∈ TC+ 𝐴 ↔ 𝑣 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
33	31	eleq2i 2853	. . . . 5 ⊢ (𝑢 ∈ TC+ 𝐴 ↔ 𝑢 ∈ ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω))
34	30, 32, 33	3imtr4g 299	. . . 4 ⊢ (𝑢 ∈ 𝑣 → (𝑣 ∈ TC+ 𝐴 → 𝑢 ∈ TC+ 𝐴))
35	34	imp 411	. . 3 ⊢ ((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
36	35	gen2 1824	. 2 ⊢ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴)
37		dftr2 5219	. 2 ⊢ (Tr TC+ 𝐴 ↔ ∀𝑢∀𝑣((𝑢 ∈ 𝑣 ∧ 𝑣 ∈ TC+ 𝐴) → 𝑢 ∈ TC+ 𝐴))
38	36, 37	mpbir 234	1 ⊢ Tr TC+ 𝐴

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  Vcvv 3453  {csn 4588  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191  Tr wtr 5217   “ cima 5664  Oncon0 6360  suc csuc 6362  Fun wfun 6530  ‘cfv 6536  ωcom 7861  reccrdg 8395  TC+ cttc 36941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 36942

This theorem is referenced by:  ttctr2  36949  ttctr3  36950  ttcss  36953  ttcel  36955  ttcidm  36958  ttciunun  36966  ttcpwss  36970  dfttc3gw  36978  ttc0elw  36982  ttc0el  36990