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Mirrors > Home > MPE Home > Th. List > nfunv | Structured version Visualization version GIF version |
Description: The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
Ref | Expression |
---|---|
nfunv | ⊢ ¬ Fun V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5300 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 0ex 4942 | . . . 4 ⊢ ∅ ∈ V | |
3 | df-rel 5273 | . . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
4 | 3 | biimpi 206 | . . . . 5 ⊢ (Rel V → V ⊆ (V × V)) |
5 | 4 | sseld 3743 | . . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V))) |
6 | 2, 5 | mpi 20 | . . 3 ⊢ (Rel V → ∅ ∈ (V × V)) |
7 | 1, 6 | mto 188 | . 2 ⊢ ¬ Rel V |
8 | funrel 6066 | . 2 ⊢ (Fun V → Rel V) | |
9 | 7, 8 | mto 188 | 1 ⊢ ¬ Fun V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ∅c0 4058 × cxp 5264 Rel wrel 5271 Fun wfun 6043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-opab 4865 df-xp 5272 df-rel 5273 df-fun 6051 |
This theorem is referenced by: (None) |
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