Theorem rankfilimbi 35257

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 482	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)))
2		limsuc 7791	. . . . . . 7 ⊢ (Lim 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ 𝐵))
3	2	ralbidv 3159	. . . . . 6 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
4	3	biimpd 229	. . . . 5 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ 𝐵))
5		fvex 6847	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ V
6	5	sucex 7751	. . . . . . 7 ⊢ suc (rank‘𝑥) ∈ V
7	6	rgenw 3055	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V
8		uniiunlem 4039	. . . . . 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 251	. . . 4 ⊢ (Lim 𝐵 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵))
11	10	impcom 407	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ⊆ 𝐵)
12	11	adantl 481	. 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 4294	. . . 4 ⊢ (Lim 𝐵 → 𝐵 ≠ ∅)
16	13, 15	jca 511	. . 3 ⊢ (Lim 𝐵 → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
17	16	ad2antll 729	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (Ord 𝐵 ∧ 𝐵 ≠ ∅))
18		rankval4b 35256	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
19	6	dfiun2 4987	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)}
20	18, 19	eqtrdi 2787	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
21	20	adantl 481	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) → (rank‘𝐴) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)})
22	21	3ad2ant1 1133	. . 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 9252	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin)
24		fissorduni 35246	. . . . 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 1163	. . . 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 1178	. . 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 2836	. 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 1373	1 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  ∪ cuni 4863  ∪ ciun 4946   “ cima 5627  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  ‘cfv 6492  Fincfn 8883  𝑅₁cr1 9674  rankcrnk 9675

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7361  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-en 8884  df-dom 8885  df-fin 8887  df-r1 9676  df-rank 9677

This theorem is referenced by:  rankfilimb  35258  r1filimi  35259