Theorem rankfilimbi 35512

Description: If all elements in 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 7847	. . . . . . 7 ⊢ (Lim 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ 𝐵))
3	2	ralbidv 3190	. . . . . 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 6898	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ V
6	5	sucex 7807	. . . . . . 7 ⊢ suc (rank‘𝑥) ∈ V
7	6	rgenw 3085	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V
8		uniiunlem 4042	. . . . . 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 6426	. . . 4 ⊢ (Lim 𝐵 → Ord 𝐵)
14		0ellim 6429	. . . . 5 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	ne0d 4295	. . . 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 35510	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
19	6	dfiun2 4998	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)}
20	18, 19	eqtrdi 2816	. . . . 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 9312	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin)
24		fissorduni 35497	. . . . 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 2865	. 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 1398	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 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ∪ cuni 4874  ∪ ciun 4958   “ cima 5666  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ‘cfv 6540  Fincfn 8945  𝑅₁cr1 9737  rankcrnk 9738

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 7738

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 7419  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-en 8946  df-dom 8947  df-fin 8949  df-r1 9739  df-rank 9740

This theorem is used by:  rankfilimb  35513  r1filimi  35514