Theorem trfr 44959

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 5623	. . . . 5 ⊢ E Se 𝐴
2		r19.21v 3159	. . . . . . 7 ⊢ (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)))
3		trpred 6307	. . . . . . . . . . 11 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → Pred( E , 𝐴, 𝑦) = 𝑦)
4		raleq 3298	. . . . . . . . . . . . 13 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On)))
5		dfss3 3938	. . . . . . . . . . . . 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 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
8	7	r1elss 9766	. . . . . . . . . . . 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 2812	. . . . . . 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 9714	. . . . 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 3159	. . . 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 3938	. . 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 2109  ∀wral 3045   ⊆ wss 3917  ∪ cuni 4874  Tr wtr 5217   E cep 5540   Fr wfr 5591   Se wse 5592   “ cima 5644  Predcpred 6276  Oncon0 6335  𝑅₁cr1 9722

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-ttrcl 9668  df-r1 9724

This theorem is referenced by:  tcfr  44960