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Mirrors > Home > ILE Home > Th. List > fun0 | 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 3321 | . 2 ⊢ ∅ ⊆ {〈∅, ∅〉} | |
2 | 0ex 3966 | . . 3 ⊢ ∅ ∈ V | |
3 | 2, 2 | funsn 5062 | . 2 ⊢ Fun {〈∅, ∅〉} |
4 | funss 5034 | . 2 ⊢ (∅ ⊆ {〈∅, ∅〉} → (Fun {〈∅, ∅〉} → Fun ∅)) | |
5 | 1, 3, 4 | mp2 16 | 1 ⊢ Fun ∅ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2999 ∅c0 3286 {csn 3446 〈cop 3449 Fun wfun 5009 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-fun 5017 |
This theorem is referenced by: fn0 5133 f10 5287 |
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