Theorem ttcwf 37130

Description: A set is well-founded iff its transitive closure is well-founded. As a corollary, the transitive closure of any well-founded set is a set. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf

Step	Hyp	Ref	Expression
1		r1rankidb 9789	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		r1tr 9761	. . . . . 6 ⊢ Tr (𝑅1‘(rank‘𝐴))
3		ttcmin 37102	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
4	1, 2, 3	sylancl 598	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
5		fvex 6895	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
6	5	elpw2 5303	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7	4, 6	sylibr 237	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
8		rankdmr1 9786	. . . . 5 ⊢ (rank‘𝐴) ∈ dom 𝑅1
9		r1sucg 9754	. . . . 5 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
10	8, 9	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
11	7, 10	eleqtrrdi 2873	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1elwf 9781	. . 3 ⊢ (TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
13	11, 12	syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
14		ttcid 37098	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
15		sswf 9793	. . 3 ⊢ ((TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ TC+ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	14, 15	mpan2 704	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
17	13, 16	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  Tr wtr 5216  dom cdm 5659   “ cima 5662  Oncon0 6361  suc csuc 6363  ‘cfv 6537  𝑅₁cr1 9747  rankcrnk 9748  TC+ cttc 37092

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-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750  df-ttc 37093

This theorem is used by:  ttcwf3  37132