Theorem ttcwf 36712

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 9717	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		r1tr 9689	. . . . . 6 ⊢ Tr (𝑅1‘(rank‘𝐴))
3		ttcmin 36684	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
4	1, 2, 3	sylancl 587	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
5		fvex 6845	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
6	5	elpw2 5269	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7	4, 6	sylibr 234	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
8		rankdmr1 9714	. . . . 5 ⊢ (rank‘𝐴) ∈ dom 𝑅1
9		r1sucg 9682	. . . . 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 2848	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1elwf 9709	. . 3 ⊢ (TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
13	11, 12	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
14		ttcid 36680	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
15		sswf 9721	. . 3 ⊢ ((TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ TC+ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	14, 15	mpan2 692	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
17	13, 16	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  Tr wtr 5193  dom cdm 5622   “ cima 5625  Oncon0 6315  suc csuc 6317  ‘cfv 6490  𝑅₁cr1 9675  rankcrnk 9676  TC+ cttc 36674

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 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-r1 9677  df-rank 9678  df-ttc 36675

This theorem is referenced by:  ttcwf3  36714