Theorem acwer1prc 35828

Description: The class of all well-orderings of the stages of the cumulative hierarchy is a proper class. (Contributed by BTernaryTau, 31-Jul-2026.)

Hypothesis

Ref	Expression
acwer1prc.1	⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}

Assertion

Ref	Expression
acwer1prc	⊢ (CHOICE → ¬ 𝑊 ∈ V)

Distinct variable group:   𝑥,𝑟

Allowed substitution hints:   𝑊(𝑥, 𝑟)

Proof of Theorem acwer1prc

Dummy variables 𝑣 𝑢 𝑦 𝑤 𝑧 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rncardr1prc 35826	. . 3 ⊢ (CHOICE → ¬ ran (card ∘ 𝑅1) ∈ V)
2		omex 9651	. . . 4 ⊢ ω ∈ V
3		difex2 7778	. . . 4 ⊢ (ω ∈ V → (ran (card ∘ 𝑅1) ∈ V ↔ (ran (card ∘ 𝑅1) ∖ ω) ∈ V))
4	2, 3	ax-mp 5	. . 3 ⊢ (ran (card ∘ 𝑅1) ∈ V ↔ (ran (card ∘ 𝑅1) ∖ ω) ∈ V)
5	1, 4	sylnib 331	. 2 ⊢ (CHOICE → ¬ (ran (card ∘ 𝑅1) ∖ ω) ∈ V)
6		simpl 488	. . . . . . . 8 ⊢ ((CHOICE ∧ 𝑦 ∈ ((card “ ran 𝑅1) ∖ ω)) → CHOICE)
7		eldifn 4067	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → ¬ 𝑦 ∈ ω)
8		eldifi 4066	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → 𝑦 ∈ (card “ ran 𝑅1))
9		imassrn 6200	. . . . . . . . . . . . . 14 ⊢ (card “ ran 𝑅1) ⊆ ran card
10	9	sseli 3905	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (card “ ran 𝑅1) → 𝑦 ∈ ran card)
11		cardf2 10031	. . . . . . . . . . . . . . 15 ⊢ card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On
12		frn 6722	. . . . . . . . . . . . . . 15 ⊢ (card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On → ran card ⊆ On)
13	11, 12	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran card ⊆ On
14	13	sseli 3905	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran card → 𝑦 ∈ On)
15	8, 10, 14	3syl 19	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → 𝑦 ∈ On)
16		onfin 9238	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (𝑦 ∈ Fin ↔ 𝑦 ∈ ω))
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → (𝑦 ∈ Fin ↔ 𝑦 ∈ ω))
18	7, 17	mtbird 328	. . . . . . . . . 10 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → ¬ 𝑦 ∈ Fin)
19		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
20		acnum 35823	. . . . . . . . . . . 12 ⊢ (CHOICE → (𝑦 ∈ V → 𝑦 ∈ dom card))
21	19, 20	mpi 21	. . . . . . . . . . 11 ⊢ (CHOICE → 𝑦 ∈ dom card)
22		infinfnum 35825	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom card → (¬ 𝑦 ∈ Fin ↔ ω ≼ 𝑦))
23	21, 22	syl 18	. . . . . . . . . 10 ⊢ (CHOICE → (¬ 𝑦 ∈ Fin ↔ ω ≼ 𝑦))
24	18, 23	imbitrid 247	. . . . . . . . 9 ⊢ (CHOICE → (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → ω ≼ 𝑦))
25	24	imp 412	. . . . . . . 8 ⊢ ((CHOICE ∧ 𝑦 ∈ ((card “ ran 𝑅1) ∖ ω)) → ω ≼ 𝑦)
26		ffun 6717	. . . . . . . . . . . 12 ⊢ (card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On → Fun card)
27	11, 26	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun card
28		fvelima 6955	. . . . . . . . . . 11 ⊢ ((Fun card ∧ 𝑦 ∈ (card “ ran 𝑅1)) → ∃𝑤 ∈ ran 𝑅1(card‘𝑤) = 𝑦)
29	27, 28	mpan 703	. . . . . . . . . 10 ⊢ (𝑦 ∈ (card “ ran 𝑅1) → ∃𝑤 ∈ ran 𝑅1(card‘𝑤) = 𝑦)
30		r1fnon 9780	. . . . . . . . . . . . . . 15 ⊢ 𝑅1 Fn On
31		fnfun 6644	. . . . . . . . . . . . . . 15 ⊢ (𝑅1 Fn On → Fun 𝑅1)
32		elrnrexdm 7094	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝑅1 → (𝑤 ∈ ran 𝑅1 → ∃𝑧 ∈ dom 𝑅1𝑤 = (𝑅1‘𝑧)))
33	30, 31, 32	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ran 𝑅1 → ∃𝑧 ∈ dom 𝑅1𝑤 = (𝑅1‘𝑧))
34	30	fndmi 6648	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅1 = On
35	34	rexeqi 3318	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ dom 𝑅1𝑤 = (𝑅1‘𝑧) ↔ ∃𝑧 ∈ On 𝑤 = (𝑅1‘𝑧))
36	33, 35	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ran 𝑅1 → ∃𝑧 ∈ On 𝑤 = (𝑅1‘𝑧))
37		rexex 3092	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ On 𝑤 = (𝑅1‘𝑧) → ∃𝑧 𝑤 = (𝑅1‘𝑧))
38	36, 37	syl 18	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran 𝑅1 → ∃𝑧 𝑤 = (𝑅1‘𝑧))
39		fveqeq2 6899	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑅1‘𝑧) → ((card‘𝑤) = 𝑦 ↔ (card‘(𝑅1‘𝑧)) = 𝑦))
40	39	biimpcd 252	. . . . . . . . . . . . 13 ⊢ ((card‘𝑤) = 𝑦 → (𝑤 = (𝑅1‘𝑧) → (card‘(𝑅1‘𝑧)) = 𝑦))
41	40	eximdv 1949	. . . . . . . . . . . 12 ⊢ ((card‘𝑤) = 𝑦 → (∃𝑧 𝑤 = (𝑅1‘𝑧) → ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦))
42	38, 41	mpan9 516	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ran 𝑅1 ∧ (card‘𝑤) = 𝑦) → ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦)
43	42	rexlimiva 3155	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ ran 𝑅1(card‘𝑤) = 𝑦 → ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦)
44	8, 29, 43	3syl 19	. . . . . . . . 9 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦)
45	44	adantl 487	. . . . . . . 8 ⊢ ((CHOICE ∧ 𝑦 ∈ ((card “ ran 𝑅1) ∖ ω)) → ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦)
46	6, 25, 45	3jca 1146	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑦 ∈ ((card “ ran 𝑅1) ∖ ω)) → (CHOICE ∧ ω ≼ 𝑦 ∧ ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦))
47		acwer1prc.1	. . . . . . . . . . 11 ⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
48	47	acwer1prclem 35827	. . . . . . . . . 10 ⊢ ((CHOICE ∧ ω ≼ 𝑦 ∧ (card‘(𝑅1‘𝑧)) = 𝑦) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦))
49	48	3expia 1139	. . . . . . . . 9 ⊢ ((CHOICE ∧ ω ≼ 𝑦) → ((card‘(𝑅1‘𝑧)) = 𝑦 → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦)))
50	49	exlimdv 1965	. . . . . . . 8 ⊢ ((CHOICE ∧ ω ≼ 𝑦) → (∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦 → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦)))
51	50	3impia 1135	. . . . . . 7 ⊢ ((CHOICE ∧ ω ≼ 𝑦 ∧ ∃𝑧(card‘(𝑅1‘𝑧)) = 𝑦) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦))
52		eleq1 2848	. . . . . . . . 9 ⊢ ((card‘𝑠) = 𝑦 → ((card‘𝑠) ∈ (card “ 𝑊) ↔ 𝑦 ∈ (card “ 𝑊)))
53	52	biimpac 484	. . . . . . . 8 ⊢ (((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦) → 𝑦 ∈ (card “ 𝑊))
54	53	exlimiv 1962	. . . . . . 7 ⊢ (∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝑦) → 𝑦 ∈ (card “ 𝑊))
55	46, 51, 54	3syl 19	. . . . . 6 ⊢ ((CHOICE ∧ 𝑦 ∈ ((card “ ran 𝑅1) ∖ ω)) → 𝑦 ∈ (card “ 𝑊))
56	55	ex 418	. . . . 5 ⊢ (CHOICE → (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → 𝑦 ∈ (card “ 𝑊)))
57	56	ssrdv 3915	. . . 4 ⊢ (CHOICE → ((card “ ran 𝑅1) ∖ ω) ⊆ (card “ 𝑊))
58		rnco2 6258	. . . . 5 ⊢ ran (card ∘ 𝑅1) = (card “ ran 𝑅1)
59	58	difeq1i 4058	. . . 4 ⊢ (ran (card ∘ 𝑅1) ∖ ω) = ((card “ ran 𝑅1) ∖ ω)
60		resima 6058	. . . 4 ⊢ ((card ↾ 𝑊) “ 𝑊) = (card “ 𝑊)
61	57, 59, 60	3sstr4g 3962	. . 3 ⊢ (CHOICE → (ran (card ∘ 𝑅1) ∖ ω) ⊆ ((card ↾ 𝑊) “ 𝑊))
62		dfac10 10223	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
63		df-fn 6546	. . . . . . . . 9 ⊢ (card Fn V ↔ (Fun card ∧ dom card = V))
64	27, 63	mpbiran 722	. . . . . . . 8 ⊢ (card Fn V ↔ dom card = V)
65	62, 64	sylbb2 241	. . . . . . 7 ⊢ (CHOICE → card Fn V)
66		dffn2 6716	. . . . . . 7 ⊢ (card Fn V ↔ card:V⟶V)
67	65, 66	sylib 221	. . . . . 6 ⊢ (CHOICE → card:V⟶V)
68		ssv 3933	. . . . . 6 ⊢ 𝑊 ⊆ V
69		fssres 6753	. . . . . 6 ⊢ ((card:V⟶V ∧ 𝑊 ⊆ V) → (card ↾ 𝑊):𝑊⟶V)
70	67, 68, 69	sylancl 598	. . . . 5 ⊢ (CHOICE → (card ↾ 𝑊):𝑊⟶V)
71		fimadmfo 6810	. . . . 5 ⊢ ((card ↾ 𝑊):𝑊⟶V → (card ↾ 𝑊):𝑊–onto→((card ↾ 𝑊) “ 𝑊))
72	70, 71	syl 18	. . . 4 ⊢ (CHOICE → (card ↾ 𝑊):𝑊–onto→((card ↾ 𝑊) “ 𝑊))
73		focdmex 7972	. . . 4 ⊢ (𝑊 ∈ V → ((card ↾ 𝑊):𝑊–onto→((card ↾ 𝑊) “ 𝑊) → ((card ↾ 𝑊) “ 𝑊) ∈ V))
74	72, 73	syl5com 32	. . 3 ⊢ (CHOICE → (𝑊 ∈ V → ((card ↾ 𝑊) “ 𝑊) ∈ V))
75		ssexg 5281	. . 3 ⊢ (((ran (card ∘ 𝑅1) ∖ ω) ⊆ ((card ↾ 𝑊) “ 𝑊) ∧ ((card ↾ 𝑊) “ 𝑊) ∈ V) → (ran (card ∘ 𝑅1) ∖ ω) ∈ V)
76	61, 74, 75	syl6an 697	. 2 ⊢ (CHOICE → (𝑊 ∈ V → (ran (card ∘ 𝑅1) ∖ ω) ∈ V))
77	5, 76	mtod 201	1 ⊢ (CHOICE → ¬ 𝑊 ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   class class class wbr 5102   We wwe 5603   × cxp 5649  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  ωcom 7881   ≈ cen 8978   ≼ cdom 8979  Fincfn 8981  𝑅₁cr1 9773  cardccrd 10023  CHOICEwac 10201

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-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

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-se 5605  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-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-rpss 7743  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-smo 8362  df-recs 8387  df-rdg 8426  df-seqom 8466  df-1o 8484  df-2o 8485  df-er 8725  df-map 8857  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-oi 9511  df-wdom 9566  df-r1 9775  df-dju 9989  df-card 10027  df-ac 10202  df-fin2 10371  df-fin4 10372  df-fin3 10373  df-fin5 10374  df-fin6 10375  df-fin7 10376

This theorem is used by:  onprcf1acwevd  35998  prconf1acwevd  35999