Theorem rankfilimbi 35133

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 7785	. . . . . . 7 ⊢ (Lim 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ 𝐵))
3	2	ralbidv 3156	. . . . . 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 6841	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ V
6	5	sucex 7745	. . . . . . 7 ⊢ suc (rank‘𝑥) ∈ V
7	6	rgenw 3052	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ V
8		uniiunlem 4036	. . . . . 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 6372	. . . 4 ⊢ (Lim 𝐵 → Ord 𝐵)
14		0ellim 6375	. . . . 5 ⊢ (Lim 𝐵 → ∅ ∈ 𝐵)
15	14	ne0d 4291	. . . 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 35132	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
19	6	dfiun2 4982	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)}
20	18, 19	eqtrdi 2784	. . . . 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 9243	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = suc (rank‘𝑥)} ∈ Fin)
24		fissorduni 35122	. . . . 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 2833	. 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 2711   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  ∪ cuni 4858  ∪ ciun 4941   “ cima 5622  Ord word 6310  Oncon0 6311  Lim wlim 6312  suc csuc 6313  ‘cfv 6486  Fincfn 8875  𝑅₁cr1 9662  rankcrnk 9663

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-en 8876  df-dom 8877  df-fin 8879  df-r1 9664  df-rank 9665

This theorem is referenced by:  rankfilimb  35134  r1filimi  35135