Theorem ttcwf 36889

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 9760	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		r1tr 9732	. . . . . 6 ⊢ Tr (𝑅1‘(rank‘𝐴))
3		ttcmin 36861	. . . . . 6 ⊢ ((𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ∧ Tr (𝑅1‘(rank‘𝐴))) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
4	1, 2, 3	sylancl 595	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
5		fvex 6880	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
6	5	elpw2 5291	. . . . 5 ⊢ (TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ TC+ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
7	4, 6	sylibr 236	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
8		rankdmr1 9757	. . . . 5 ⊢ (rank‘𝐴) ∈ dom 𝑅1
9		r1sucg 9725	. . . . 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 2874	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
12		r1elwf 9752	. . 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 36857	. . 3 ⊢ 𝐴 ⊆ TC+ 𝐴
15		sswf 9764	. . 3 ⊢ ((TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ TC+ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	14, 15	mpan2 701	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
17	13, 16	impbii 211	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 208   = wceq 1561   ∈ wcel 2143   ⊆ wss 3905  𝒫 cpw 4556  ∪ cuni 4866  Tr wtr 5208  dom cdm 5648   “ cima 5651  Oncon0 6346  suc csuc 6348  ‘cfv 6521  𝑅₁cr1 9718  rankcrnk 9719  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-r1 9720  df-rank 9721  df-ttc 36852

This theorem is referenced by:  ttcwf3  36891