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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 4199 | . 2 ⊢ ∅ ⊆ {〈∅, ∅〉} | |
2 | 0ex 5016 | . . 3 ⊢ ∅ ∈ V | |
3 | 2, 2 | funsn 6179 | . 2 ⊢ Fun {〈∅, ∅〉} |
4 | funss 6146 | . 2 ⊢ (∅ ⊆ {〈∅, ∅〉} → (Fun {〈∅, ∅〉} → Fun ∅)) | |
5 | 1, 3, 4 | mp2 9 | 1 ⊢ Fun ∅ |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3798 ∅c0 4146 {csn 4399 〈cop 4405 Fun wfun 6121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-fun 6129 |
This theorem is referenced by: funcnv0 6192 fn0 6248 f10 6414 0fsupp 8572 strle1 16339 lubfun 17340 glbfun 17353 1pthdlem1 27507 |
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