Theorem ncanth 2946

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by nvelv 1483). Specifically, the identity function maps the universe onto its power class. Compare canth 2945 that works for sets. See also the remark in ru 1437 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure, since in NF the universal class is a set.

Assertion

Ref	Expression
ncanth	⊢ I:V–onto→℘V

Proof of Theorem ncanth

Step	Hyp	Ref	Expression
1		f1ovi 2826	. . 3 ⊢ I:V–1-1-onto→V
2		pwv 1884	. . . 4 ⊢ ℘V = V
3		f1oeq3 2797	. . . 4 ⊢ (℘V = V → (I:V–1-1-onto→℘V ↔ I:V–1-1-onto→V))
4	2, 3	ax-mp 6	. . 3 ⊢ (I:V–1-1-onto→℘V ↔ I:V–1-1-onto→V)
5	1, 4	mpbir 165	. 2 ⊢ I:V–1-1-onto→℘V
6		f1ofo 2806	. 2 ⊢ (I:V–1-1-onto→℘V → I:V–onto→℘V)
7	5, 6	ax-mp 6	1 ⊢ I:V–onto→℘V

Syntax hints:   ↔ wb 127   = wceq 1091  Vcvv 1348  ℘cpw 1798  Icid 2057  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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