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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 4639 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 0ex 4116 | . . . 4 ⊢ ∅ ∈ V | |
3 | df-rel 4618 | . . . . . 6 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
4 | 3 | biimpi 119 | . . . . 5 ⊢ (Rel V → V ⊆ (V × V)) |
5 | 4 | sseld 3146 | . . . 4 ⊢ (Rel V → (∅ ∈ V → ∅ ∈ (V × V))) |
6 | 2, 5 | mpi 15 | . . 3 ⊢ (Rel V → ∅ ∈ (V × V)) |
7 | 1, 6 | mto 657 | . 2 ⊢ ¬ Rel V |
8 | funrel 5215 | . 2 ⊢ (Fun V → Rel V) | |
9 | 7, 8 | mto 657 | 1 ⊢ ¬ Fun V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 ∅c0 3414 × cxp 4609 Rel wrel 4616 Fun wfun 5192 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051 df-xp 4617 df-rel 4618 df-fun 5200 |
This theorem is referenced by: (None) |
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