Theorem r1filimi 9906

Description: If all elements of a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

Ref	Expression
r1filimi	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem r1filimi

Dummy variables 𝑤 𝑎 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3316	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
2		eleq1 2848	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
3	1, 2	imbi12d 347	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
4	3	imbi2d 343	. . . . 5 ⊢ (𝑎 = 𝐴 → ((Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On))) ↔ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)))))
5		r1fun 9774	. . . . . . . . 9 ⊢ Fun 𝑅1
6		eluniima 7254	. . . . . . . . 9 ⊢ (Fun 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦)))
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦))
8		limord 6424	. . . . . . . . . . . 12 ⊢ (Lim 𝐵 → Ord 𝐵)
9		ordsson 7797	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
10	8, 9	syl 18	. . . . . . . . . . 11 ⊢ (Lim 𝐵 → 𝐵 ⊆ On)
11	10	sseld 3907	. . . . . . . . . 10 ⊢ (Lim 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
12	11	anim1d 623	. . . . . . . . 9 ⊢ (Lim 𝐵 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦))))
13	12	reximdv2 3172	. . . . . . . 8 ⊢ (Lim 𝐵 → (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
14	7, 13	biimtrid 245	. . . . . . 7 ⊢ (Lim 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
15	14	ralimdv 3176	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦)))
16		vex 3454	. . . . . . . 8 ⊢ 𝑎 ∈ V
17	16	tz9.12 9800	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
18		eluniima 7254	. . . . . . . 8 ⊢ (Fun 𝑅1 → (𝑎 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦)))
19	5, 18	ax-mp 5	. . . . . . 7 ⊢ (𝑎 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑎 ∈ (𝑅1‘𝑦))
20	17, 19	sylibr 237	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑎 ∈ ∪ (𝑅1 “ On))
21	15, 20	syl6 36	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝑎 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝑎 ∈ ∪ (𝑅1 “ On)))
22	4, 21	vtoclg 3517	. . . 4 ⊢ (𝐴 ∈ Fin → (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))))
23	22	impcomd 417	. . 3 ⊢ (𝐴 ∈ Fin → ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)))
24	23	3impib 1134	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
25		simp3 1156	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → Lim 𝐵)
26		simp1 1154	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ Fin)
27		eluniima 7254	. . . . . . . . 9 ⊢ (Fun 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧)))
28	5, 27	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧))
29		df-rex 3087	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)))
30		rankr1ai 9809	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝑧)
31		ordtr1 6407	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ 𝑧 ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
32	30, 31	sylani 616	. . . . . . . . . . 11 ⊢ (Ord 𝐵 → ((𝑥 ∈ (𝑅1‘𝑧) ∧ 𝑧 ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
33	32	ancomsd 471	. . . . . . . . . 10 ⊢ (Ord 𝐵 → ((𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
34	33	exlimdv 1965	. . . . . . . . 9 ⊢ (Ord 𝐵 → (∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑥 ∈ (𝑅1‘𝑧)) → (rank‘𝑥) ∈ 𝐵))
35	29, 34	biimtrid 245	. . . . . . . 8 ⊢ (Ord 𝐵 → (∃𝑧 ∈ 𝐵 𝑥 ∈ (𝑅1‘𝑧) → (rank‘𝑥) ∈ 𝐵))
36	28, 35	biimtrid 245	. . . . . . 7 ⊢ (Ord 𝐵 → (𝑥 ∈ ∪ (𝑅1 “ 𝐵) → (rank‘𝑥) ∈ 𝐵))
37	36	ralimdv 3176	. . . . . 6 ⊢ (Ord 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
38	8, 37	syl 18	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
39	38	impcom 413	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
40	39	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵)
41		rankfilimbi 9905	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
42	26, 24, 40, 25, 41	syl22anc 852	. 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
43		fveq2 6885	. . . . 5 ⊢ (𝑤 = suc (rank‘𝐴) → (𝑅1‘𝑤) = (𝑅1‘suc (rank‘𝐴)))
44	43	eleq2d 2846	. . . 4 ⊢ (𝑤 = suc (rank‘𝐴) → (𝐴 ∈ (𝑅1‘𝑤) ↔ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))))
45		limsuc 7860	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
46	45	biimpa 482	. . . . 5 ⊢ ((Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
47	46	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → suc (rank‘𝐴) ∈ 𝐵)
48		rankidb 9811	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
49	48	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
50	44, 47, 49	rspcedvdw 3579	. . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
51		eluniima 7254	. . . 4 ⊢ (Fun 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤)))
52	5, 51	ax-mp 5	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∃𝑤 ∈ 𝐵 𝐴 ∈ (𝑅1‘𝑤))
53	50, 52	sylibr 237	. 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵 ∧ (rank‘𝐴) ∈ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
54	24, 25, 42, 53	syl3anc 1398	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ∪ cuni 4866   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  Fincfn 8976  𝑅₁cr1 9769  rankcrnk 9770

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-en 8977  df-dom 8978  df-fin 8980  df-r1 9771  df-rank 9772

This theorem is used by:  elhf4  9915  r1filim  35781