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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 3452 | . 2 ⊢ ∅ ⊆ {〈∅, ∅〉} | |
2 | 0ex 4114 | . . 3 ⊢ ∅ ∈ V | |
3 | 2, 2 | funsn 5244 | . 2 ⊢ Fun {〈∅, ∅〉} |
4 | funss 5215 | . 2 ⊢ (∅ ⊆ {〈∅, ∅〉} → (Fun {〈∅, ∅〉} → Fun ∅)) | |
5 | 1, 3, 4 | mp2 16 | 1 ⊢ Fun ∅ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3121 ∅c0 3414 {csn 3581 〈cop 3584 Fun wfun 5190 |
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 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-fun 5198 |
This theorem is referenced by: fn0 5315 f10 5474 ennnfonelemj0 12345 |
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