Theorem trssfir1omregs 35606

Description: If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfir1om 35565 replaces setinds2 9727 with setinds2regs 35601. (Contributed by BTernaryTau, 20-Jan-2026.)

Assertion

Ref	Expression
trssfir1omregs	⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))

Proof of Theorem trssfir1omregs

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

Step	Hyp	Ref	Expression
1		eleq1w 2848	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	3anbi1d 1468	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) ↔ (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin)))
3		eleq1w 2848	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	2, 3	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)) ↔ ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω))))
5		ssel2 3933	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ Fin)
6	5	ancoms 464	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ Fin)
7	6	3adant2 1149	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ Fin)
8	7	a1i 11	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ Fin))
9		trel 5228	. . . . . . . . . . . . 13 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
10	9	expcomd 422	. . . . . . . . . . . 12 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
11	10	impcom 413	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
12	11	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
13		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → Tr 𝐴)
14	13	a1d 26	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → Tr 𝐴))
15		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
16	15	a1d 26	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → 𝐴 ⊆ Fin))
17	12, 14, 16	3jcad 1147	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin)))
18	17	ralrimiv 3158	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin))
19		ralim 3107	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
20	18, 19	syl5 35	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
21	8, 20	jcad 522	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω))))
22		r1omhf 35558	. . . . . 6 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
23	21, 22	imbitrrdi 255	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)))
24	4, 23	setinds2regs 35601	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω))
25	24	3expib 1140	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)))
26	25	com12 33	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ ω)))
27	26	ssrdv 3944	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  ∪ cuni 4874  Tr wtr 5220   “ cima 5666  ωcom 7868  Fincfn 8949  𝑅₁cr1 9741

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 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-regs 35596

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-en 8950  df-dom 8951  df-fin 8953  df-r1 9743  df-rank 9744

This theorem is used by:  r1omhfbregs  35607