Theorem tcfr 45705

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 9858	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
2		r1elssi 9780	. . 3 ⊢ ((TC‘𝐴) ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
3		wffr 45703	. . . 4 ⊢ E Fr ∪ (𝑅1 “ On)
4		frss 5627	. . . 4 ⊢ ((TC‘𝐴) ⊆ ∪ (𝑅1 “ On) → ( E Fr ∪ (𝑅1 “ On) → E Fr (TC‘𝐴)))
5	3, 4	mpi 21	. . 3 ⊢ ((TC‘𝐴) ⊆ ∪ (𝑅1 “ On) → E Fr (TC‘𝐴))
6	1, 2, 5	3syl 19	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → E Fr (TC‘𝐴))
7		tcfr.1	. . . . 5 ⊢ 𝐴 ∈ V
8		tcid 9709	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ (TC‘𝐴)
10		tctr 9710	. . . . 5 ⊢ Tr (TC‘𝐴)
11		trfr 45704	. . . . 5 ⊢ ((Tr (TC‘𝐴) ∧ E Fr (TC‘𝐴)) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
12	10, 11	mpan 703	. . . 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 9781	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))
15	13, 14	sylibr 237	. 2 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	6, 15	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  ∪ cuni 4874  Tr wtr 5220   E cep 5562   Fr wfr 5613   “ cima 5666  Oncon0 6364  ‘cfv 6540  TCctc 9706  𝑅₁cr1 9737

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-ttrcl 9680  df-tc 9707  df-r1 9739  df-rank 9740  df-relp 45685

This theorem is used by: (None)