Theorem rankfilimb 35111

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 9717	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
2	1	3ad2ant2 1134	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
3		limord 6367	. . . . . . 7 ⊢ (Lim 𝐵 → Ord 𝐵)
4		ordtr1 6350	. . . . . . 7 ⊢ (Ord 𝐵 → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
5	3, 4	syl 17	. . . . . 6 ⊢ (Lim 𝐵 → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
6	5	3ad2ant3 1135	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
7	2, 6	syland 603	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((𝑥 ∈ 𝐴 ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
8	7	expcomd 416	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ 𝐵)))
9	8	ralrimdv 3130	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
10		rankfilimbi 35110	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
11	10	3impb 1114	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
12	11	3com23 1126	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵 ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵)
13	12	3expia 1121	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵) → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → (rank‘𝐴) ∈ 𝐵))
14	13	3impa 1109	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → (rank‘𝐴) ∈ 𝐵))
15	9, 14	impbid 212	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2111  ∀wral 3047  ∪ cuni 4859   “ cima 5619  Ord word 6305  Oncon0 6306  Lim wlim 6307  ‘cfv 6481  Fincfn 8869  𝑅₁cr1 9655  rankcrnk 9656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-en 8870  df-dom 8871  df-fin 8873  df-r1 9657  df-rank 9658

This theorem is referenced by: (None)