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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 4562 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 0ex 4050 | . . . 4 ⊢ ∅ ∈ V | |
3 | df-rel 4541 | . . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
4 | 3 | biimpi 119 | . . . . 5 ⊢ (Rel V → V ⊆ (V × V)) |
5 | 4 | sseld 3091 | . . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V))) |
6 | 2, 5 | mpi 15 | . . 3 ⊢ (Rel V → ∅ ∈ (V × V)) |
7 | 1, 6 | mto 651 | . 2 ⊢ ¬ Rel V |
8 | funrel 5135 | . 2 ⊢ (Fun V → Rel V) | |
9 | 7, 8 | mto 651 | 1 ⊢ ¬ Fun V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 ∅c0 3358 × cxp 4532 Rel wrel 4539 Fun wfun 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-rel 4541 df-fun 5120 |
This theorem is referenced by: (None) |
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