Theorem fineqvnttrclselem1 35453

Description: Lemma for fineqvnttrclse 35456. (Contributed by BTernaryTau, 12-Jan-2026.)

Assertion

Ref	Expression
fineqvnttrclselem1	⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)

Distinct variable groups:   𝐴,𝑑   𝐵,𝑑

Proof of Theorem fineqvnttrclselem1

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4093	. . 3 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		eleq1 2857	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑑) = 𝐵 → ((𝐴 +o 𝑑) ∈ ω ↔ 𝐵 ∈ ω))
3	2	biimparc 484	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
4	3	adantll 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
5	4	3adant2 1147	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
6		nnarcl 8598	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
7	6	adantlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
8	7	3adant3 1148	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
9	5, 8	mpbid 235	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 ∈ ω ∧ 𝑑 ∈ ω))
10	9	simprd 500	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → 𝑑 ∈ ω)
11	10	rabssdv 4036	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω)
12		nnon 7864	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
13		oawordeu 8536	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
14		reusn 4695	. . . . . . . . 9 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤})
15		snfi 9036	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
16		eleq1 2857	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin ↔ {𝑤} ∈ Fin))
17	15, 16	mpbiri 261	. . . . . . . . . 10 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
18	17	exlimiv 1957	. . . . . . . . 9 ⊢ (∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
19	14, 18	sylbi 220	. . . . . . . 8 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
20	13, 19	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
21	12, 20	sylanl2 693	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
22		nnunifi 9247	. . . . . 6 ⊢ (({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω ∧ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
23	11, 21, 22	syl2an2r 697	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
24		oawordex 8538	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
25	12, 24	sylan2 604	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
26	25	notbid 321	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
27	26	biimpa 481	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
28		ralnex 3097	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
29		rabeq0 4351	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅ ↔ ∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵)
30	29	biimpri 231	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
31	30	unieqd 4886	. . . . . . . . 9 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ ∅)
32		uni0 4902	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
33	31, 32	eqtrdi 2820	. . . . . . . 8 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
34		peano1 7881	. . . . . . . 8 ⊢ ∅ ∈ ω
35	33, 34	eqeltrdi 2877	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
36	28, 35	sylbir 238	. . . . . 6 ⊢ (¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
37	27, 36	syl 18	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
38	23, 37	pm2.61dan 824	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
39	38	expcom 418	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
40	1, 39	syl 18	. 2 ⊢ (𝐵 ∈ (ω ∖ 1o) → (𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
41		simpl 487	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → 𝐴 ∈ On)
42		df-oadd 8453	. . . . . . . 8 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
43	42	mpondm0 7648	. . . . . . 7 ⊢ (¬ (𝐴 ∈ On ∧ 𝑑 ∈ On) → (𝐴 +o 𝑑) = ∅)
44	41, 43	nsyl5 160	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (𝐴 +o 𝑑) = ∅)
45		eldifsnneq 4760	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ {∅}) → ¬ 𝐵 = ∅)
46		df1o2 8456	. . . . . . . 8 ⊢ 1o = {∅}
47	46	difeq2i 4086	. . . . . . 7 ⊢ (ω ∖ 1o) = (ω ∖ {∅})
48	45, 47	eleq2s 2887	. . . . . 6 ⊢ (𝐵 ∈ (ω ∖ 1o) → ¬ 𝐵 = ∅)
49		eqtr2 2790	. . . . . . 7 ⊢ (((𝐴 +o 𝑑) = 𝐵 ∧ (𝐴 +o 𝑑) = ∅) → 𝐵 = ∅)
50	49	stoic1b 1800	. . . . . 6 ⊢ (((𝐴 +o 𝑑) = ∅ ∧ ¬ 𝐵 = ∅) → ¬ (𝐴 +o 𝑑) = 𝐵)
51	44, 48, 50	syl2anr 608	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ¬ (𝐴 +o 𝑑) = 𝐵)
52	51	ralrimivw 3167	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵)
53	52, 35	syl 18	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
54	53	ex 417	. 2 ⊢ (𝐵 ∈ (ω ∖ 1o) → (¬ 𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
55	40, 54	pm2.61d 181	1 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  {csn 4591  ∪ cuni 4873   ↦ cmpt 5193  Oncon0 6358  suc csuc 6360  ‘cfv 6534  (class class class)co 7408  ωcom 7858  reccrdg 8392  1_oc1o 8442   +_o coa 8446  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-en 8940  df-fin 8943

This theorem is referenced by:  fineqvnttrclselem2  35454  fineqvnttrclselem3  35455  fineqvnttrclse  35456