Theorem trssfir1om 35286

Description: If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.)

Assertion

Ref	Expression
trssfir1om	⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))

Proof of Theorem trssfir1om

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

Step	Hyp	Ref	Expression
1		eleq1w 2820	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	3anbi1d 1443	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) ↔ (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin)))
3		eleq1w 2820	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ 𝑦 ∈ ∪ (𝑅1 “ ω)))
4	2, 3	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)) ↔ ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω))))
5		ssel2 3930	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ Fin)
6	5	ancoms 458	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ Fin)
7	6	3adant2 1132	. . . . . . . 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 5215	. . . . . . . . . . . . 13 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
10	9	expcomd 416	. . . . . . . . . . . 12 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
11	10	impcom 407	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
12	11	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
13		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → Tr 𝐴)
14	13	a1d 25	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → Tr 𝐴))
15		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
16	15	a1d 25	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → 𝐴 ⊆ Fin))
17	12, 14, 16	3jcad 1130	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin)))
18	17	ralrimiv 3129	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin))
19		ralim 3078	. . . . . . . 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 34	. . . . . . 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 512	. . . . . 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 35281	. . . . . 6 ⊢ (𝑥 ∈ ∪ (𝑅1 “ ω) ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝑅1 “ ω)))
23	21, 22	imbitrrdi 252	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ((𝑦 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑦 ∈ ∪ (𝑅1 “ ω)) → ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)))
24	4, 23	setinds2 9672	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω))
25	24	3expib 1123	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝑥 ∈ ∪ (𝑅1 “ ω)))
26	25	com12 32	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ ω)))
27	26	ssrdv 3941	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  ∪ cuni 4865  Tr wtr 5207   “ cima 5635  ωcom 7818  Fincfn 8895  𝑅₁cr1 9686

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562

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 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-en 8896  df-dom 8897  df-fin 8899  df-r1 9688  df-rank 9689

This theorem is referenced by:  r1omhfb  35287