Theorem ncanth 3914

Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by nvelv 2718). Specifically, the identity function maps the universe onto its power class. Compare canth 3913 that works for sets. See also the remark in ru 1941 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set.

Assertion

Ref	Expression
ncanth	⊢ I:V–onto→℘V

Proof of Theorem ncanth

Step	Hyp	Ref	Expression
1		f1ovi 3724	. . 3 ⊢ I:V–1-1-onto→V
2		pwv 2506	. . . 4 ⊢ ℘V = V
3		f1oeq3 3692	. . . 4 ⊢ (℘V = V → (I:V–1-1-onto→℘V ↔ I:V–1-1-onto→V))
4	2, 3	ax-mp 7	. . 3 ⊢ (I:V–1-1-onto→℘V ↔ I:V–1-1-onto→V)
5	1, 4	mpbir 190	. 2 ⊢ I:V–1-1-onto→℘V
6		f1ofo 3701	. 2 ⊢ (I:V–1-1-onto→℘V → I:V–onto→℘V)
7	5, 6	ax-mp 7	1 ⊢ I:V–onto→℘V

Syntax hints:   ↔ wb 146   = wceq 958  Vcvv 1814  ℘cpw 2405  Icid 2837  –onto→wfo 3186  –1-1-onto→wf1o 3187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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