Theorem ttcwf 36706

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 9728	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		r1tr 9700	. . . . . 6 ⊢ Tr (𝑅1‘(rank‘𝐴))
3		ttcmin 36678	. . . . . 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 6853	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
6	5	elpw2 5275	. . . . 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 9725	. . . . 5 ⊢ (rank‘𝐴) ∈ dom 𝑅1
9		r1sucg 9693	. . . . 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 2847	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1elwf 9720	. . 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 36674	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
15		sswf 9732	. . 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 3889  𝒫 cpw 4541  ∪ cuni 4850  Tr wtr 5192  dom cdm 5631   “ cima 5634  Oncon0 6323  suc csuc 6325  ‘cfv 6498  𝑅₁cr1 9686  rankcrnk 9687  TC+ cttc 36668

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-rank 9689  df-ttc 36669

This theorem is referenced by:  ttcwf3  36708