Theorem rankfilimbi 35289

Description: If all elements in a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

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

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Proof of Theorem rankfilimbi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)))
2		limsuc 7796	. . . . . . 7 ⊢ (Lim 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ 𝐵))
3	2	ralbidv 3163	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
4	3	biimpd 230	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
5		fvex 6847	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ V
6	5	sucex 7756	. . . . . . 7 ⊢ suc (rank‘𝑥) ∈ V
7	6	rgenw 3058	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V
8		uniiunlem 4025	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V → (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵))
9	7, 8	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵 ↔ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
10	4, 9	imbitrdi 252	. . . 4 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵))
11	10	impcom 408	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
12	11	adantl 482	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
13		limord 6378	. . . 4 ⊢ (Lim 𝐵 → Ord 𝐵)
14		0ellim 6381	. . . . 5 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	ne0d 4277	. . . 4 ⊢ (Lim 𝐵 → 𝐵 ≠ ∅)
16	13, 15	jca 516	. . 3 ⊢ (Lim 𝐵 → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
17	16	ad2antll 735	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
18		rankval4b 35288	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
19	6	dfiun2 4968	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)}
20	18, 19	eqtrdi 2791	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
21	20	adantl 482	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
22	21	3ad2ant1 1139	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
23		abrexfi 9259	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin)
24		fissorduni 35278	. . . . 5 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
25	23, 24	syl3an1 1169	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
26	25	3adant1r 1184	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ 𝐵)
27	22, 26	eqeltrd 2840	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → (rank‘𝐴) ∈ 𝐵)
28	1, 12, 17, 27	syl3anc 1379	1 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {cab 2718   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  ∪ cuni 4845  ∪ ciun 4928   “ cima 5628  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  ‘cfv 6492  Fincfn 8890  𝑅₁cr1 9684  rankcrnk 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-en 8891  df-dom 8892  df-fin 8894  df-r1 9686  df-rank 9687

This theorem is referenced by:  rankfilimb  35290  r1filimi  35291