Theorem rankfilimb 35474

Description: The rank of a finite well-founded set is less than a limit ordinal iff the ranks of all of its elements are less than that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rankfilimb

Step	Hyp	Ref	Expression
1		rankelb 9797	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
2	1	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
3		limord 6424	. . . . . . 7 ⊢ (Lim 𝐵 → Ord 𝐵)
4		ordtr1 6407	. . . . . . 7 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
5	3, 4	syl 18	. . . . . 6 ⊢ (Lim 𝐵 → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
6	5	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
7	2, 6	syland 614	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((𝑥 ∈ 𝐴 ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
8	7	expcomd 421	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ 𝐵)))
9	8	ralrimdv 3163	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
10		rankfilimbi 35473	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
11	10	3impb 1132	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
12	11	3com23 1144	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵 ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵)
13	12	3expia 1139	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵) → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → (rank‘𝐴) ∈ 𝐵))
14	13	3impa 1127	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → (rank‘𝐴) ∈ 𝐵))
15	9, 14	impbid 215	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079  ∪ cuni 4873   “ cima 5666  Ord word 6361  Oncon0 6362  Lim wlim 6363  ‘cfv 6538  Fincfn 8944  𝑅₁cr1 9735  rankcrnk 9736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  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 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 7415  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-en 8945  df-dom 8946  df-fin 8948  df-r1 9737  df-rank 9738

This theorem is referenced by: (None)