Theorem trfr 44945

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 5613	. . . . 5 ⊢ E Se 𝐴
2		r19.21v 3158	. . . . . . 7 ⊢ (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)))
3		trpred 6292	. . . . . . . . . . 11 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → Pred( E , 𝐴, 𝑦) = 𝑦)
4		raleq 3293	. . . . . . . . . . . . 13 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On)))
5		dfss3 3932	. . . . . . . . . . . . 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 3448	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
8	7	r1elss 9735	. . . . . . . . . . . 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 2811	. . . . . . 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 9683	. . . . 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 3158	. . . 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 3932	. . 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 3044   ⊆ wss 3911  ∪ cuni 4867  Tr wtr 5209   E cep 5530   Fr wfr 5581   Se wse 5582   “ cima 5634  Predcpred 6261  Oncon0 6320  𝑅₁cr1 9691

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-ttrcl 9637  df-r1 9693

This theorem is referenced by:  tcfr  44946