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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 4405 | . 2 ⊢ ∅ ⊆ {〈∅, ∅〉} | |
2 | 0ex 5312 | . . 3 ⊢ ∅ ∈ V | |
3 | 2, 2 | funsn 6620 | . 2 ⊢ Fun {〈∅, ∅〉} |
4 | funss 6586 | . 2 ⊢ (∅ ⊆ {〈∅, ∅〉} → (Fun {〈∅, ∅〉} → Fun ∅)) | |
5 | 1, 3, 4 | mp2 9 | 1 ⊢ Fun ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3962 ∅c0 4338 {csn 4630 〈cop 4636 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-fun 6564 |
This theorem is referenced by: funcnv0 6633 fn0 6699 0fsupp 9427 strle1 17191 lubfun 18409 glbfun 18422 1pthdlem1 30163 fineqvac 35089 |
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