Theorem trfr 44935

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 5636	. . . . 5 ⊢ E Se 𝐴
2		r19.21v 3165	. . . . . . 7 ⊢ (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)))
3		trpred 6320	. . . . . . . . . . 11 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → Pred( E , 𝐴, 𝑦) = 𝑦)
4		raleq 3302	. . . . . . . . . . . . 13 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On)))
5		dfss3 3947	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On))
6	4, 5	bitr4di 289	. . . . . . . . . . . 12 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ⊆ ∪ (𝑅1 “ On)))
7		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
8	7	r1elss 9818	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ⊆ ∪ (𝑅1 “ On))
9	6, 8	bitr4di 289	. . . . . . . . . . 11 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
10	3, 9	syl 17	. . . . . . . . . 10 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
11	10	biimpd 229	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ ∪ (𝑅1 “ On)))
12	11	expcom 413	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (Tr 𝐴 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ ∪ (𝑅1 “ On))))
13	12	a2d 29	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)) → (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On))))
14	2, 13	biimtrid 242	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) → (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On))))
15		eleq1w 2817	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ∈ ∪ (𝑅1 “ On)))
16	15	imbi2d 340	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On))))
17	14, 16	frins2 9766	. . . . 5 ⊢ (( E Fr 𝐴 ∧ E Se 𝐴) → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
18	1, 17	mpan2 691	. . . 4 ⊢ ( E Fr 𝐴 → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
19		r19.21v 3165	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)))
20	18, 19	sylib 218	. . 3 ⊢ ( E Fr 𝐴 → (Tr 𝐴 → ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On)))
21		dfss3 3947	. . 3 ⊢ (𝐴 ⊆ ∪ (𝑅1 “ On) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ ∪ (𝑅1 “ On))
22	20, 21	imbitrrdi 252	. 2 ⊢ ( E Fr 𝐴 → (Tr 𝐴 → 𝐴 ⊆ ∪ (𝑅1 “ On)))
23	22	impcom 407	1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ∪ cuni 4883  Tr wtr 5229   E cep 5552   Fr wfr 5603   Se wse 5604   “ cima 5657  Predcpred 6289  Oncon0 6352  𝑅₁cr1 9774

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-ttrcl 9720  df-r1 9776

This theorem is referenced by:  tcfr  44936