Theorem rncardr1prc 35826

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 7796	. 2 ⊢ ¬ On ∈ V
2		dfac10 10223	. . . . . 6 ⊢ (CHOICE ↔ dom card = V)
3		df-card 10027	. . . . . . . 8 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
4	3	funmpt2 6584	. . . . . . 7 ⊢ Fun card
5		df-fn 6546	. . . . . . 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 9789	. . . . . 6 ⊢ 𝑅1:On–1-1→V
9		f1f 6783	. . . . . 6 ⊢ (𝑅1:On–1-1→V → 𝑅1:On⟶V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 𝑅1:On⟶V
11		fnfco 6752	. . . . 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 6727	. . . 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 10678	. . . 4 ⊢ Smo (card ∘ 𝑅1)
16		smo11 8380	. . . 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 7973	. . . 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 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450  ∩ cint 4906   class class class wbr 5102  dom cdm 5651  ran crn 5652   ∘ ccom 5655  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  Smo wsmo 8361   ≈ cen 8978  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

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-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-smo 8362  df-recs 8387  df-rdg 8426  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-r1 9775  df-card 10027  df-ac 10202

This theorem is used by:  acwer1prc  35828