Theorem rncardr1prc 35891

Description: The Axiom of Choice implies that the cardinalities of the layers of the cumulative hierarchy form a proper class. (Contributed by BTernaryTau, 2-Jul-2026.)

Assertion

Ref	Expression
rncardr1prc	⊢ (CHOICE → ¬ ran (card ∘ 𝑅1) ∈ V)

Proof of Theorem rncardr1prc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onprc 7824	. 2 ⊢ ¬ On ∈ V
2		dfac10 10255	. . . . . 6 ⊢ (CHOICE ↔ dom card = V)
3		df-card 10057	. . . . . . . 8 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	funmpt2 6607	. . . . . . 7 ⊢ Fun card
5		df-fn 6569	. . . . . . 7 ⊢ (card Fn V ↔ (Fun card ∧ dom card = V))
6	4, 5	mpbiran 722	. . . . . 6 ⊢ (card Fn V ↔ dom card = V)
7	2, 6	sylbb2 241	. . . . 5 ⊢ (CHOICE → card Fn V)
8		r111 9819	. . . . . 6 ⊢ 𝑅1:On⟶1-1V
9		f1f 6808	. . . . . 6 ⊢ (𝑅1:On⟶1-1V → 𝑅1:On⟶V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 𝑅1:On⟶V
11		fnfco 6777	. . . . 5 ⊢ ((card Fn V ∧ 𝑅1:On⟶V) → (card ∘ 𝑅1) Fn On)
12	7, 10, 11	sylancl 598	. . . 4 ⊢ (CHOICE → (card ∘ 𝑅1) Fn On)
13		dffn3 6752	. . . 4 ⊢ ((card ∘ 𝑅1) Fn On ↔ (card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1))
14	12, 13	sylib 221	. . 3 ⊢ (CHOICE → (card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1))
15		smobeth 10710	. . . 4 ⊢ Smo (card ∘ 𝑅1)
16		smo11 8410	. . . 4 ⊢ (((card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1) ∧ Smo (card ∘ 𝑅1)) → (card ∘ 𝑅1):On⟶1-1ran (card ∘ 𝑅1))
17	15, 16	mpan2 704	. . 3 ⊢ ((card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1) → (card ∘ 𝑅1):On⟶1-1ran (card ∘ 𝑅1))
18		f1dmex 8003	. . . 4 ⊢ (((card ∘ 𝑅1):On⟶1-1ran (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ∈ V) → On ∈ V)
19	18	ex 418	. . 3 ⊢ ((card ∘ 𝑅1):On⟶1-1ran (card ∘ 𝑅1) → (ran (card ∘ 𝑅1) ∈ V → On ∈ V))
20	14, 17, 19	3syl 19	. 2 ⊢ (CHOICE → (ran (card ∘ 𝑅1) ∈ V → On ∈ V))
21	1, 20	mtoi 202	1 ⊢ (CHOICE → ¬ ran (card ∘ 𝑅1) ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  ∩ cint 4912   class class class wbr 5108  dom cdm 5658  ran crn 5659   ∘ ccom 5662  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  Smo wsmo 8391   ≈ cen 9008  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

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-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-smo 8392  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-r1 9805  df-card 10057  df-ac 10234

This theorem is used by:  acwer1prc  35893