Theorem fineqvnttrclselem1 35277

Description: Lemma for fineqvnttrclse 35280. (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 4083	. . 3 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		eleq1 2824	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑑) = 𝐵 → ((𝐴 +o 𝑑) ∈ ω ↔ 𝐵 ∈ ω))
3	2	biimparc 479	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
4	3	adantll 714	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
5	4	3adant2 1131	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
6		nnarcl 8544	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
7	6	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
8	7	3adant3 1132	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
9	5, 8	mpbid 232	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 ∈ ω ∧ 𝑑 ∈ ω))
10	9	simprd 495	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → 𝑑 ∈ ω)
11	10	rabssdv 4026	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω)
12		nnon 7814	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
13		oawordeu 8482	. . . . . . . 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 8980	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
16		eleq1 2824	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin ↔ {𝑤} ∈ Fin))
17	15, 16	mpbiri 258	. . . . . . . . . 10 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
18	17	exlimiv 1931	. . . . . . . . 9 ⊢ (∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤} → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
19	14, 18	sylbi 217	. . . . . . . 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 681	. . . . . 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 685	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
24		oawordex 8484	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
25	12, 24	sylan2 593	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
26	25	notbid 318	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (¬ 𝐴 ⊆ 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
27	26	biimpa 476	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
28		ralnex 3062	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
29		rabeq0 4340	. . . . . . . . . . 11 ⊢ ({𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅ ↔ ∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵)
30	29	biimpri 228	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
31	30	unieqd 4876	. . . . . . . . 9 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ ∅)
32		uni0 4891	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
33	31, 32	eqtrdi 2787	. . . . . . . 8 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
34		peano1 7831	. . . . . . . 8 ⊢ ∅ ∈ ω
35	33, 34	eqeltrdi 2844	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
36	28, 35	sylbir 235	. . . . . 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 812	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
39	38	expcom 413	. . 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 482	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → 𝐴 ∈ On)
42		df-oadd 8401	. . . . . . . 8 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
43	42	mpondm0 7598	. . . . . . 7 ⊢ (¬ (𝐴 ∈ On ∧ 𝑑 ∈ On) → (𝐴 +o 𝑑) = ∅)
44	41, 43	nsyl5 159	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (𝐴 +o 𝑑) = ∅)
45		eldifsnneq 4747	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ {∅}) → ¬ 𝐵 = ∅)
46		df1o2 8404	. . . . . . . 8 ⊢ 1o = {∅}
47	46	difeq2i 4075	. . . . . . 7 ⊢ (ω ∖ 1o) = (ω ∖ {∅})
48	45, 47	eleq2s 2854	. . . . . 6 ⊢ (𝐵 ∈ (ω ∖ 1o) → ¬ 𝐵 = ∅)
49		eqtr2 2757	. . . . . . 7 ⊢ (((𝐴 +o 𝑑) = 𝐵 ∧ (𝐴 +o 𝑑) = ∅) → 𝐵 = ∅)
50	49	stoic1b 1774	. . . . . 6 ⊢ (((𝐴 +o 𝑑) = ∅ ∧ ¬ 𝐵 = ∅) → ¬ (𝐴 +o 𝑑) = 𝐵)
51	44, 48, 50	syl2anr 597	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ¬ (𝐴 +o 𝑑) = 𝐵)
52	51	ralrimivw 3132	. . . 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 412	. 2 ⊢ (𝐵 ∈ (ω ∖ 1o) → (¬ 𝐴 ∈ On → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω))
55	40, 54	pm2.61d 179	1 ⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ∃!wreu 3348  {crab 3399  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  {csn 4580  ∪ cuni 4863   ↦ cmpt 5179  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808  reccrdg 8340  1_oc1o 8390   +_o coa 8394  Fincfn 8883

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-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-rmo 3350  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-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-en 8884  df-fin 8887

This theorem is referenced by:  fineqvnttrclselem2  35278  fineqvnttrclselem3  35279  fineqvnttrclse  35280