Theorem fineqvnttrclselem1 35857

Description: Lemma for fineqvnttrclse 35860. (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 4066	. . 3 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		eleq1 2848	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑑) = 𝐵 → ((𝐴 +o 𝑑) ∈ ω ↔ 𝐵 ∈ ω))
3	2	biimparc 485	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
4	3	adantll 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
5	4	3adant2 1149	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
6		nnarcl 8633	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
7	6	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
8	7	3adant3 1150	. . . . . . . . 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 501	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → 𝑑 ∈ ω)
11	10	rabssdv 3998	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω)
12		nnon 7887	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
13		oawordeu 8571	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
14		reusn 4684	. . . . . . . . 9 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤})
15		snfi 9079	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
16		eleq1 2848	. . . . . . . . . . 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 1962	. . . . . . . . 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 694	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
22		nnunifi 9291	. . . . . 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 698	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
24		oawordex 8573	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
25	12, 24	sylan2 605	. . . . . . . 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 482	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
28		ralnex 3088	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
29		rabeq0 4337	. . . . . . . . . . 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 4879	. . . . . . . . 9 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ ∅)
32		uni0 4895	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
33	31, 32	eqtrdi 2811	. . . . . . . 8 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
34		peano1 7904	. . . . . . . 8 ⊢ ∅ ∈ ω
35	33, 34	eqeltrdi 2868	. . . . . . 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 825	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
39	38	expcom 419	. . 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 488	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → 𝐴 ∈ On)
42		df-oadd 8488	. . . . . . . 8 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
43	42	mpondm0 7665	. . . . . . 7 ⊢ (¬ (𝐴 ∈ On ∧ 𝑑 ∈ On) → (𝐴 +o 𝑑) = ∅)
44	41, 43	nsyl5 160	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (𝐴 +o 𝑑) = ∅)
45		eldifsnneq 4750	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ {∅}) → ¬ 𝐵 = ∅)
46		df1o2 8491	. . . . . . . 8 ⊢ 1o = {∅}
47	46	difeq2i 4059	. . . . . . 7 ⊢ (ω ∖ 1o) = (ω ∖ {∅})
48	45, 47	eleq2s 2878	. . . . . 6 ⊢ (𝐵 ∈ (ω ∖ 1o) → ¬ 𝐵 = ∅)
49		eqtr2 2781	. . . . . . 7 ⊢ (((𝐴 +o 𝑑) = 𝐵 ∧ (𝐴 +o 𝑑) = ∅) → 𝐵 = ∅)
50	49	stoic1b 1805	. . . . . 6 ⊢ (((𝐴 +o 𝑑) = ∅ ∧ ¬ 𝐵 = ∅) → ¬ (𝐴 +o 𝑑) = 𝐵)
51	44, 48, 50	syl2anr 609	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ¬ (𝐴 +o 𝑑) = 𝐵)
52	51	ralrimivw 3158	. . . 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 418	. 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044  ∅c0 4277  {csn 4584  ∪ cuni 4866   ↦ cmpt 5185  Oncon0 6365  suc csuc 6367  ‘cfv 6543  (class class class)co 7424  ωcom 7881  reccrdg 8425  1_oc1o 8477   +_o coa 8481  Fincfn 8981

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-oadd 8488  df-en 8982  df-fin 8985

This theorem is used by:  fineqvnttrclselem2  35858  fineqvnttrclselem3  35859  fineqvnttrclse  35860