Theorem rankfilimbi 9939

Description: If all elements of a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

Ref	Expression
rankfilimbi	⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Proof of Theorem rankfilimbi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)))
2		limsuc 7894	. . . . . . 7 ⊢ (Lim 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ 𝐵))
3	2	ralbidv 3189	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
4	3	biimpd 232	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
5		fvex 6929	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ V
6	5	sucex 7854	. . . . . . 7 ⊢ suc (rank‘𝑥) ∈ V
7	6	rgenw 3084	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V
8		uniiunlem 4015	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V → (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵))
9	7, 8	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
10	4, 9	imbitrdi 254	. . . 4 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵))
11	10	impcom 413	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
12	11	adantl 487	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
13		limord 6450	. . . 4 ⊢ (Lim 𝐵 → Ord 𝐵)
14		0ellim 6453	. . . . 5 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	ne0d 4291	. . . 4 ⊢ (Lim 𝐵 → 𝐵 ≠ ∅)
16	13, 15	jca 521	. . 3 ⊢ (Lim 𝐵 → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
17	16	ad2antll 742	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
18		rankval4b 9917	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
19	6	dfiun2 4995	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)}
20	18, 19	eqtrdi 2814	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
21	20	adantl 487	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
22	21	3ad2ant1 1151	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
23		abrexfi 9379	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin)
24		fissorduni 9320	. . . . 5 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
25	23, 24	syl3an1 1181	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
26	25	3adant1r 1196	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
27	22, 26	eqeltrd 2864	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → (rank‘𝐴) ∈ 𝐵)
28	1, 12, 17, 27	syl3anc 1399	1 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4872  ∪ ciun 4956   “ cima 5661  Ord word 6387  Oncon0 6388  Lim wlim 6389  suc csuc 6390  ‘cfv 6566  Fincfn 9011  𝑅₁cr1 9803  rankcrnk 9804

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-en 9012  df-dom 9013  df-fin 9015  df-r1 9805  df-rank 9806

This theorem is used by:  r1filimi  9940  rankfilimb  35853