Theorem trfr 45239

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 5607	. . . . 5 ⊢ E Se 𝐴
2		r19.21v 3162	. . . . . . 7 ⊢ (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)(Tr 𝐴 → 𝑧 ∈ ∪ (𝑅1 “ On)) ↔ (Tr 𝐴 → ∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On)))
3		trpred 6290	. . . . . . . . . . 11 ⊢ ((Tr 𝐴 ∧ 𝑦 ∈ 𝐴) → Pred( E , 𝐴, 𝑦) = 𝑦)
4		raleq 3294	. . . . . . . . . . . . 13 ⊢ (Pred( E , 𝐴, 𝑦) = 𝑦 → (∀𝑧 ∈ Pred ( E , 𝐴, 𝑦)𝑧 ∈ ∪ (𝑅1 “ On) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ ∪ (𝑅1 “ On)))
5		dfss3 3923	. . . . . . . . . . . . 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 3445	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
8	7	r1elss 9722	. . . . . . . . . . . 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 2820	. . . . . . 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 9670	. . . . 5 ⊢ (( E Fr 𝐴 ∧ E Se 𝐴) → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
18	1, 17	mpan2 692	. . . 4 ⊢ ( E Fr 𝐴 → ∀𝑦 ∈ 𝐴 (Tr 𝐴 → 𝑦 ∈ ∪ (𝑅1 “ On)))
19		r19.21v 3162	. . . 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 3923	. . 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 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  ∪ cuni 4864  Tr wtr 5206   E cep 5524   Fr wfr 5575   Se wse 5576   “ cima 5628  Predcpred 6259  Oncon0 6318  𝑅₁cr1 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-ttrcl 9621  df-r1 9680

This theorem is referenced by:  tcfr  45240