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Mirrors > Home > ILE Home > Th. List > nfunv | 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 4654 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
3 | df-rel 4633 | . . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
4 | 3 | biimpi 120 | . . . . 5 ⊢ (Rel V → V ⊆ (V × V)) |
5 | 4 | sseld 3154 | . . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V))) |
6 | 2, 5 | mpi 15 | . . 3 ⊢ (Rel V → ∅ ∈ (V × V)) |
7 | 1, 6 | mto 662 | . 2 ⊢ ¬ Rel V |
8 | funrel 5233 | . 2 ⊢ (Fun V → Rel V) | |
9 | 7, 8 | mto 662 | 1 ⊢ ¬ Fun V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 ∅c0 3422 × cxp 4624 Rel wrel 4631 Fun wfun 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 df-xp 4632 df-rel 4633 df-fun 5218 |
This theorem is referenced by: (None) |
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