Theorem rankfilimb 35717

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 9826	. . . . . 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 6423	. . . . . . 7 ⊢ (Lim 𝐵 → Ord 𝐵)
4		ordtr1 6406	. . . . . . 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 3161	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
10		rankfilimbi 9895	. . . . . 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 2145  ∀wral 3077  ∪ cuni 4867   “ cima 5654  Ord word 6360  Oncon0 6361  Lim wlim 6362  ‘cfv 6537  Fincfn 8966  𝑅₁cr1 9759  rankcrnk 9760

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-en 8967  df-dom 8968  df-fin 8970  df-r1 9761  df-rank 9762

This theorem is used by: (None)