Theorem fineqvnttrclselem1 35299

Description: Lemma for fineqvnttrclse 35302. (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 4085	. . 3 ⊢ (𝐵 ∈ (ω ∖ 1o) → 𝐵 ∈ ω)
2		eleq1 2825	. . . . . . . . . . . 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 715	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
5	4	3adant2 1132	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On ∧ (𝐴 +o 𝑑) = 𝐵) → (𝐴 +o 𝑑) ∈ ω)
6		nnarcl 8554	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
7	6	adantlr 716	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝑑 ∈ On) → ((𝐴 +o 𝑑) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝑑 ∈ ω)))
8	7	3adant3 1133	. . . . . . . . 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 4028	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ⊆ ω)
12		nnon 7824	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
13		oawordeu 8492	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
14		reusn 4686	. . . . . . . . 9 ⊢ (∃!𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵 ↔ ∃𝑤{𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = {𝑤})
15		snfi 8992	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
16		eleq1 2825	. . . . . . . . . . 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 1932	. . . . . . . . 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 682	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ Fin)
22		nnunifi 9203	. . . . . 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 686	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
24		oawordex 8494	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵))
25	12, 24	sylan2 594	. . . . . . . 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 3064	. . . . . . 7 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 ↔ ¬ ∃𝑑 ∈ On (𝐴 +o 𝑑) = 𝐵)
29		rabeq0 4342	. . . . . . . . . . 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 4878	. . . . . . . . 9 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∪ ∅)
32		uni0 4893	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
33	31, 32	eqtrdi 2788	. . . . . . . 8 ⊢ (∀𝑑 ∈ On ¬ (𝐴 +o 𝑑) = 𝐵 → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} = ∅)
34		peano1 7841	. . . . . . . 8 ⊢ ∅ ∈ ω
35	33, 34	eqeltrdi 2845	. . . . . . 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 813	. . . 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 8411	. . . . . . . 8 ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
43	42	mpondm0 7608	. . . . . . 7 ⊢ (¬ (𝐴 ∈ On ∧ 𝑑 ∈ On) → (𝐴 +o 𝑑) = ∅)
44	41, 43	nsyl5 159	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (𝐴 +o 𝑑) = ∅)
45		eldifsnneq 4749	. . . . . . 7 ⊢ (𝐵 ∈ (ω ∖ {∅}) → ¬ 𝐵 = ∅)
46		df1o2 8414	. . . . . . . 8 ⊢ 1o = {∅}
47	46	difeq2i 4077	. . . . . . 7 ⊢ (ω ∖ 1o) = (ω ∖ {∅})
48	45, 47	eleq2s 2855	. . . . . 6 ⊢ (𝐵 ∈ (ω ∖ 1o) → ¬ 𝐵 = ∅)
49		eqtr2 2758	. . . . . . 7 ⊢ (((𝐴 +o 𝑑) = 𝐵 ∧ (𝐴 +o 𝑑) = ∅) → 𝐵 = ∅)
50	49	stoic1b 1775	. . . . . 6 ⊢ (((𝐴 +o 𝑑) = ∅ ∧ ¬ 𝐵 = ∅) → ¬ (𝐴 +o 𝑑) = 𝐵)
51	44, 48, 50	syl2anr 598	. . . . 5 ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ ¬ 𝐴 ∈ On) → ¬ (𝐴 +o 𝑑) = 𝐵)
52	51	ralrimivw 3134	. . . 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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  {crab 3401  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  Oncon0 6325  suc csuc 6327  ‘cfv 6500  (class class class)co 7368  ωcom 7818  reccrdg 8350  1_oc1o 8400   +_o coa 8404  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-en 8896  df-fin 8899

This theorem is referenced by:  fineqvnttrclselem2  35300  fineqvnttrclselem3  35301  fineqvnttrclse  35302