Theorem rncardr1prc 35758

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 7792	. 2 ⊢ ¬ On ∈ V
2		dfac10 10216	. . . . . 6 ⊢ (CHOICE ↔ dom card = V)
3		df-card 10020	. . . . . . . 8 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	funmpt2 6579	. . . . . . 7 ⊢ Fun card
5		df-fn 6541	. . . . . . 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 9782	. . . . . 6 ⊢ 𝑅1:On–1-1→V
9		f1f 6778	. . . . . 6 ⊢ (𝑅1:On–1-1→V → 𝑅1:On⟶V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 𝑅1:On⟶V
11		fnfco 6747	. . . . 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 6722	. . . 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 10671	. . . 4 ⊢ Smo (card ∘ 𝑅1)
16		smo11 8372	. . . 4 ⊢ (((card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1) ∧ Smo (card ∘ 𝑅1)) → (card ∘ 𝑅1):On–1-1→ran (card ∘ 𝑅1))
17	15, 16	mpan2 704	. . 3 ⊢ ((card ∘ 𝑅1):On⟶ran (card ∘ 𝑅1) → (card ∘ 𝑅1):On–1-1→ran (card ∘ 𝑅1))
18		f1dmex 7969	. . . 4 ⊢ (((card ∘ 𝑅1):On–1-1→ran (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ∈ V) → On ∈ V)
19	18	ex 418	. . 3 ⊢ ((card ∘ 𝑅1):On–1-1→ran (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 2145  {crab 3413  Vcvv 3451  ∩ cint 4907   class class class wbr 5103  dom cdm 5651  ran crn 5652   ∘ ccom 5655  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  Smo wsmo 8353   ≈ cen 8970  𝑅₁cr1 9766  cardccrd 10016  CHOICEwac 10194

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-smo 8354  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-r1 9768  df-card 10020  df-ac 10195

This theorem is used by:  acwer1prc  35760