Theorem rankfilimb 35277

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 9748	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
2	1	3ad2ant2 1135	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
3		limord 6386	. . . . . . 7 ⊢ (Lim 𝐵 → Ord 𝐵)
4		ordtr1 6369	. . . . . . 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 1136	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → (((rank‘𝑥) ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ 𝐵) → (rank‘𝑥) ∈ 𝐵))
7	2, 6	syland 604	. . . 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 3136	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
10		rankfilimbi 35276	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
11	10	3impb 1115	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → (rank‘𝐴) ∈ 𝐵)
12	11	3com23 1127	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵 ∧ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵)
13	12	3expia 1122	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ Lim 𝐵) → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → (rank‘𝐴) ∈ 𝐵))
14	13	3impa 1110	. 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 1087   ∈ wcel 2114  ∀wral 3052  ∪ cuni 4865   “ cima 5635  Ord word 6324  Oncon0 6325  Lim wlim 6326  ‘cfv 6500  Fincfn 8895  𝑅₁cr1 9686  rankcrnk 9687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-en 8896  df-dom 8897  df-fin 8899  df-r1 9688  df-rank 9689

This theorem is referenced by: (None)