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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 4614 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 0ex 4091 | . . . 4 ⊢ ∅ ∈ V | |
3 | df-rel 4593 | . . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
4 | 3 | biimpi 119 | . . . . 5 ⊢ (Rel V → V ⊆ (V × V)) |
5 | 4 | sseld 3127 | . . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V))) |
6 | 2, 5 | mpi 15 | . . 3 ⊢ (Rel V → ∅ ∈ (V × V)) |
7 | 1, 6 | mto 652 | . 2 ⊢ ¬ Rel V |
8 | funrel 5187 | . 2 ⊢ (Fun V → Rel V) | |
9 | 7, 8 | mto 652 | 1 ⊢ ¬ Fun V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2128 Vcvv 2712 ⊆ wss 3102 ∅c0 3394 × cxp 4584 Rel wrel 4591 Fun wfun 5164 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-opab 4026 df-xp 4592 df-rel 4593 df-fun 5172 |
This theorem is referenced by: (None) |
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