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| Mirrors > Home > MPE Home > Th. List > fun0 | Structured version Visualization version GIF version | ||
| Description: The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
| Ref | Expression |
|---|---|
| fun0 | ⊢ Fun ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ss 4400 | . 2 ⊢ ∅ ⊆ {〈∅, ∅〉} | |
| 2 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 3 | 2, 2 | funsn 6619 | . 2 ⊢ Fun {〈∅, ∅〉} |
| 4 | funss 6585 | . 2 ⊢ (∅ ⊆ {〈∅, ∅〉} → (Fun {〈∅, ∅〉} → Fun ∅)) | |
| 5 | 1, 3, 4 | mp2 9 | 1 ⊢ Fun ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: funcnv0 6632 fn0 6699 0fsupp 9430 strle1 17195 lubfun 18397 glbfun 18410 1pthdlem1 30154 fineqvac 35111 |
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