Theorem tcfr 45414

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 9805	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
2		r1elssi 9727	. . 3 ⊢ ((TC‘𝐴) ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
3		wffr 45412	. . . 4 ⊢ E Fr ∪ (𝑅1 “ On)
4		frss 5589	. . . 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 9656	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ (TC‘𝐴)
10		tctr 9657	. . . . 5 ⊢ Tr (TC‘𝐴)
11		trfr 45413	. . . . 5 ⊢ ((Tr (TC‘𝐴) ∧ E Fr (TC‘𝐴)) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
12	10, 11	mpan 696	. . . 4 ⊢ ( E Fr (TC‘𝐴) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
13	9, 12	sstrid 3933	. . 3 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
14	7	r1elss 9728	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))
15	13, 14	sylibr 235	. 2 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	6, 15	impbii 210	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))

Syntax hints:   ↔ wb 207   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890  ∪ cuni 4845  Tr wtr 5186   E cep 5524   Fr wfr 5575   “ cima 5628  Oncon0 6317  ‘cfv 6492  TCctc 9653  𝑅₁cr1 9684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-ttrcl 9627  df-tc 9654  df-r1 9686  df-rank 9687  df-relp 45394

This theorem is referenced by: (None)