Theorem tcfr 44980

Description: A set is well-founded if and only if its transitive closure is well-founded by ∈. This characterization of well-founded sets is that in Definition I.9.20 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)

Hypothesis

Ref	Expression
tcfr.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tcfr	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))

Proof of Theorem tcfr

Step	Hyp	Ref	Expression
1		tcwf 9923	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
2		r1elssi 9845	. . 3 ⊢ ((TC‘𝐴) ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
3		wffr 44978	. . . 4 ⊢ E Fr ∪ (𝑅1 “ On)
4		frss 5649	. . . 4 ⊢ ((TC‘𝐴) ⊆ ∪ (𝑅1 “ On) → ( E Fr ∪ (𝑅1 “ On) → E Fr (TC‘𝐴)))
5	3, 4	mpi 20	. . 3 ⊢ ((TC‘𝐴) ⊆ ∪ (𝑅1 “ On) → E Fr (TC‘𝐴))
6	1, 2, 5	3syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → E Fr (TC‘𝐴))
7		tcfr.1	. . . . 5 ⊢ 𝐴 ∈ V
8		tcid 9779	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ (TC‘𝐴)
10		tctr 9780	. . . . 5 ⊢ Tr (TC‘𝐴)
11		trfr 44979	. . . . 5 ⊢ ((Tr (TC‘𝐴) ∧ E Fr (TC‘𝐴)) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
12	10, 11	mpan 690	. . . 4 ⊢ ( E Fr (TC‘𝐴) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
13	9, 12	sstrid 3995	. . 3 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
14	7	r1elss 9846	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))
15	13, 14	sylibr 234	. 2 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	6, 15	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951  ∪ cuni 4907  Tr wtr 5259   E cep 5583   Fr wfr 5634   “ cima 5688  Oncon0 6384  ‘cfv 6561  TCctc 9776  𝑅₁cr1 9802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-ttrcl 9748  df-tc 9777  df-r1 9804  df-rank 9805  df-relp 44964

This theorem is referenced by: (None)