Theorem acwer1prc 35893

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 35891	. . 3 ⊢ (CHOICE → ¬ ran (card ∘ 𝑅1) ∈ V)
2		omex 9681	. . . 4 ⊢ ω ∈ V
3		difex2 7806	. . . 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 4071	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → ¬ 𝑦 ∈ ω)
8		eldifi 4070	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → 𝑦 ∈ (card “ ran 𝑅1))
9		imassrn 6216	. . . . . . . . . . . . . 14 ⊢ (card “ ran 𝑅1) ⊆ ran card
10	9	sseli 3909	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (card “ ran 𝑅1) → 𝑦 ∈ ran card)
11		cardf2 10061	. . . . . . . . . . . . . . 15 ⊢ card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On
12		frn 6747	. . . . . . . . . . . . . . 15 ⊢ (card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On → ran card ⊆ On)
13	11, 12	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran card ⊆ On
14	13	sseli 3909	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran card → 𝑦 ∈ On)
15	8, 10, 14	3syl 19	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((card “ ran 𝑅1) ∖ ω) → 𝑦 ∈ On)
16		onfin 9268	. . . . . . . . . . . 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 3458	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
20		acnum 35888	. . . . . . . . . . . 12 ⊢ (CHOICE → (𝑦 ∈ V → 𝑦 ∈ dom card))
21	19, 20	mpi 21	. . . . . . . . . . 11 ⊢ (CHOICE → 𝑦 ∈ dom card)
22		infinfnum 35890	. . . . . . . . . . 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 6742	. . . . . . . . . . . 12 ⊢ (card:{𝑢 ∣ ∃𝑣 ∈ On 𝑣 ≈ 𝑢}⟶On → Fun card)
27	11, 26	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun card
28		fvelima 6981	. . . . . . . . . . 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 9810	. . . . . . . . . . . . . . 15 ⊢ 𝑅1 Fn On
31		fnfun 6669	. . . . . . . . . . . . . . 15 ⊢ (𝑅1 Fn On → Fun 𝑅1)
32		elrnrexdm 7121	. . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . 15 ⊢ dom 𝑅1 = On
35	34	rexeqi 3322	. . . . . . . . . . . . . 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 3096	. . . . . . . . . . . . 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 6925	. . . . . . . . . . . . . 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 1952	. . . . . . . . . . . 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 3159	. . . . . . . . . 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 35892	. . . . . . . . . 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 1968	. . . . . . . 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 2852	. . . . . . . . 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 1965	. . . . . . 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 3919	. . . 4 ⊢ (CHOICE → ((card “ ran 𝑅1) ∖ ω) ⊆ (card “ 𝑊))
58		rnco2 6274	. . . . 5 ⊢ ran (card ∘ 𝑅1) = (card “ ran 𝑅1)
59	58	difeq1i 4062	. . . 4 ⊢ (ran (card ∘ 𝑅1) ∖ ω) = ((card “ ran 𝑅1) ∖ ω)
60		resima 6074	. . . 4 ⊢ ((card ↾ 𝑊) “ 𝑊) = (card “ 𝑊)
61	57, 59, 60	3sstr4g 3966	. . 3 ⊢ (CHOICE → (ran (card ∘ 𝑅1) ∖ ω) ⊆ ((card ↾ 𝑊) “ 𝑊))
62		dfac10 10255	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
63		df-fn 6569	. . . . . . . . 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 6741	. . . . . . 7 ⊢ (card Fn V ↔ card:V⟶V)
67	65, 66	sylib 221	. . . . . 6 ⊢ (CHOICE → card:V⟶V)
68		ssv 3937	. . . . . 6 ⊢ 𝑊 ⊆ V
69		fssres 6778	. . . . . 6 ⊢ ((card:V⟶V ∧ 𝑊 ⊆ V) → (card ↾ 𝑊):𝑊⟶V)
70	67, 68, 69	sylancl 598	. . . . 5 ⊢ (CHOICE → (card ↾ 𝑊):𝑊⟶V)
71		fimadmfo 6835	. . . . 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 8002	. . . 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 5288	. . 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 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   class class class wbr 5108   We wwe 5610   × cxp 5656  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661   ∘ ccom 5662  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  ωcom 7911   ≈ cen 9008   ≼ cdom 9009  Fincfn 9011  𝑅₁cr1 9803  cardccrd 10053  CHOICEwac 10233

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-rpss 7771  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-smo 8392  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-wdom 9596  df-r1 9805  df-dju 10019  df-card 10057  df-ac 10234  df-fin2 10403  df-fin4 10404  df-fin3 10405  df-fin5 10406  df-fin6 10407  df-fin7 10408

This theorem is used by:  onprcf1acwevd  36088  prconf1acwevd  36089