Theorem tcfr 45499

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 9834	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
2		r1elssi 9756	. . 3 ⊢ ((TC‘𝐴) ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
3		wffr 45497	. . . 4 ⊢ E Fr ∪ (𝑅1 “ On)
4		frss 5607	. . . 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 9685	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (TC‘𝐴))
9	7, 8	ax-mp 5	. . . 4 ⊢ 𝐴 ⊆ (TC‘𝐴)
10		tctr 9686	. . . . 5 ⊢ Tr (TC‘𝐴)
11		trfr 45498	. . . . 5 ⊢ ((Tr (TC‘𝐴) ∧ E Fr (TC‘𝐴)) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
12	10, 11	mpan 700	. . . 4 ⊢ ( E Fr (TC‘𝐴) → (TC‘𝐴) ⊆ ∪ (𝑅1 “ On))
13	9, 12	sstrid 3945	. . 3 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
14	7	r1elss 9757	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On))
15	13, 14	sylibr 236	. 2 ⊢ ( E Fr (TC‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
16	6, 15	impbii 211	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))

Syntax hints:   ↔ wb 208   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4862  Tr wtr 5204   E cep 5542   Fr wfr 5593   “ cima 5646  Oncon0 6340  ‘cfv 6515  TCctc 9682  𝑅₁cr1 9713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-inf2 9589

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-oadd 8434  df-ttrcl 9656  df-tc 9683  df-r1 9715  df-rank 9716  df-relp 45479

This theorem is referenced by: (None)