Theorem trfr 45712

Description: A transitive class well-founded by ∈ is a subclass of the class of well-founded sets. Part of Lemma I.9.21 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)

Assertion

Ref	Expression
trfr	⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))

Proof of Theorem trfr

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epse 5648	. . . . 5 ⊢ E Se 𝐴
2		r19.21v 3193	. . . . . . 7 ⊢ (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)))
3		trpred 6339	. . . . . . . . . . 11 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → Pred( E , 𝐴, 𝑦) = 𝑦)
4		raleq 3323	. . . . . . . . . . . . 13 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On)))
5		dfss3 3929	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On))
6	4, 5	bitr4di 292	. . . . . . . . . . . 12 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ⊆ ∪ (𝑅1 “ On)))
7		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
8	7	r1elss 9788	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ⊆ ∪ (𝑅1 “ On))
9	6, 8	bitr4di 292	. . . . . . . . . . 11 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
10	3, 9	syl 18	. . . . . . . . . 10 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
11	10	biimpd 232	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ ∪ (𝑅1 “ On)))
12	11	expcom 419	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (Tr 𝐴 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ ∪ (𝑅1 “ On))))
13	12	a2d 30	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)) → (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On))))
14	2, 13	biimtrid 245	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) → (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On))))
15		eleq1w 2849	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ∈ ∪ (𝑅1 “ On)))
16	15	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On))))
17	14, 16	frins2 9736	. . . . 5 ⊢ (( E Fr 𝐴 ∧ E Se 𝐴) → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
18	1, 17	mpan2 704	. . . 4 ⊢ ( E Fr 𝐴 → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
19		r19.21v 3193	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)))
20	18, 19	sylib 221	. . 3 ⊢ ( E Fr 𝐴 → (Tr 𝐴 → ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)))
21		dfss3 3929	. . 3 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On))
22	20, 21	imbitrrdi 255	. 2 ⊢ ( E Fr 𝐴 → (Tr 𝐴 → 𝐴 ⊆ ∪ (𝑅1 “ On)))
23	22	impcom 413	1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908  ∪ cuni 4877  Tr wtr 5223   E cep 5565   Fr wfr 5616   Se wse 5617   “ cima 5669  Predcpred 6308  Oncon0 6367  𝑅₁cr1 9744

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 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-ttrcl 9687  df-r1 9746

This theorem is used by:  tcfr  45713