Theorem fineqvnttrclselem1 35302

Description: Lemma for fineqvnttrclse 35305. (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 4061	. . 3 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		eleq1 2827	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑑) = 𝐵 → ((𝐴 +o 𝑑) ∈ ω ↔ 𝐵 ∈ ω))
3	2	biimparc 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
4	3	adantll 720	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
5	4	3adant2 1137	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
6		nnarcl 8542	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
7	6	adantlr 721	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
8	7	3adant3 1138	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
9	5, 8	mpbid 233	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 ∈ ω ∧ 𝑑 ∈ ω))
10	9	simprd 496	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → 𝑑 ∈ ω)
11	10	rabssdv 4005	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω)
12		nnon 7812	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
13		oawordeu 8480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
14		reusn 4659	. . . . . . . . 9 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤})
15		snfi 8980	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
16		eleq1 2827	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin ↔ {𝑤} ∈ Fin))
17	15, 16	mpbiri 259	. . . . . . . . . 10 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
18	17	exlimiv 1937	. . . . . . . . 9 ⊢ (∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
19	14, 18	sylbi 218	. . . . . . . 8 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
20	13, 19	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
21	12, 20	sylanl2 687	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
22		nnunifi 9191	. . . . . 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 691	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
24		oawordex 8482	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
25	12, 24	sylan2 599	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
26	25	notbid 319	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
27	26	biimpa 477	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
28		ralnex 3065	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
29		rabeq0 4316	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅ ↔ ∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵)
30	29	biimpri 229	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
31	30	unieqd 4851	. . . . . . . . 9 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ ∅)
32		uni0 4866	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
33	31, 32	eqtrdi 2790	. . . . . . . 8 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
34		peano1 7829	. . . . . . . 8 ⊢ ∅ ∈ ω
35	33, 34	eqeltrdi 2847	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
36	28, 35	sylbir 236	. . . . . 6 ⊢ (¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
37	27, 36	syl 17	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
38	23, 37	pm2.61dan 818	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
39	38	expcom 414	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
40	1, 39	syl 17	. 2 ⊢ (𝐵 ∈ (ω ∖ 1o) → (𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
41		simpl 483	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → 𝐴 ∈ On)
42		df-oadd 8399	. . . . . . . 8 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
43	42	mpondm0 7596	. . . . . . 7 ⊢ (¬ (𝐴 ∈ On ∧ 𝑑 ∈ On) → (𝐴 +o 𝑑) = ∅)
44	41, 43	nsyl5 159	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (𝐴 +o 𝑑) = ∅)
45		eldifsnneq 4724	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ {∅}) → ¬ 𝐵 = ∅)
46		df1o2 8402	. . . . . . . 8 ⊢ 1o = {∅}
47	46	difeq2i 4054	. . . . . . 7 ⊢ (ω ∖ 1o) = (ω ∖ {∅})
48	45, 47	eleq2s 2857	. . . . . 6 ⊢ (𝐵 ∈ (ω ∖ 1o) → ¬ 𝐵 = ∅)
49		eqtr2 2760	. . . . . . 7 ⊢ (((𝐴 +o 𝑑) = 𝐵 ∧ (𝐴 +o 𝑑) = ∅) → 𝐵 = ∅)
50	49	stoic1b 1780	. . . . . 6 ⊢ (((𝐴 +o 𝑑) = ∅ ∧ ¬ 𝐵 = ∅) → ¬ (𝐴 +o 𝑑) = 𝐵)
51	44, 48, 50	syl2anr 603	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ¬ (𝐴 +o 𝑑) = 𝐵)
52	51	ralrimivw 3135	. . . 4 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵)
53	52, 35	syl 17	. . 3 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
54	53	ex 413	. 2 ⊢ (𝐵 ∈ (ω ∖ 1o) → (¬ 𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
55	40, 54	pm2.61d 180	1 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  ∃!wreu 3342  {crab 3391  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  {csn 4555  ∪ cuni 4838   ↦ cmpt 5153  Oncon0 6310  suc csuc 6312  ‘cfv 6485  (class class class)co 7356  ωcom 7806  reccrdg 8338  1_oc1o 8388   +_o coa 8392  Fincfn 8883

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 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

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 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-en 8884  df-fin 8887

This theorem is referenced by:  fineqvnttrclselem2  35303  fineqvnttrclselem3  35304  fineqvnttrclse  35305