Theorem tcfr 44937

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 9798	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
2		r1elssi 9720	. . 3 ⊢ ((TC‘𝐴) ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
3		wffr 44935	. . . 4 ⊢ E Fr ∪ (𝑅1 “ On)
4		frss 5587	. . . 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 9654	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ (TC‘𝐴)
10		tctr 9655	. . . . 5 ⊢ Tr (TC‘𝐴)
11		trfr 44936	. . . . 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 3949	. . 3 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
14	7	r1elss 9721	. . 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 2109  Vcvv 3438   ⊆ wss 3905  ∪ cuni 4861  Tr wtr 5202   E cep 5522   Fr wfr 5573   “ cima 5626  Oncon0 6311  ‘cfv 6486  TCctc 9651  𝑅₁cr1 9677

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-ttrcl 9623  df-tc 9652  df-r1 9679  df-rank 9680  df-relp 44917

This theorem is referenced by: (None)