Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ncanth | Structured version Visualization version GIF version |
Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4724). Specifically, the identity function maps the universe onto its power class. Compare canth 6508 that works for sets. See also the remark in ru 3401 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. (Contributed by NM, 29-Jun-2004.) |
Ref | Expression |
---|---|
ncanth | ⊢ I :V–onto→𝒫 V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ovi 6087 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | pwv 4371 | . . . 4 ⊢ 𝒫 V = V | |
3 | f1oeq3 6042 | . . . 4 ⊢ (𝒫 V = V → ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V) |
5 | 1, 4 | mpbir 220 | . 2 ⊢ I :V–1-1-onto→𝒫 V |
6 | f1ofo 6057 | . 2 ⊢ ( I :V–1-1-onto→𝒫 V → I :V–onto→𝒫 V) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 Vcvv 3173 𝒫 cpw 4108 I cid 4948 –onto→wfo 5802 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |