Theorem rankfilimb 35781

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 9837	. . . . . 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 615	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((𝑥 ∈ 𝐴 ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
8	7	expcomd 422	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ 𝐵)))
9	8	ralrimdv 3160	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
10		rankfilimbi 9906	. . . . . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  ∀wral 3076  ∪ cuni 4866   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  ‘cfv 6539  Fincfn 8977  𝑅₁cr1 9770  rankcrnk 9771

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 7752

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 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-en 8978  df-dom 8979  df-fin 8981  df-r1 9772  df-rank 9773

This theorem is used by: (None)