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Mirrors > Home > ILE Home > Th. List > smo0 | GIF version |
Description: The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Ref | Expression |
---|---|
smo0 | ⊢ Smo ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ord0 4308 | . . 3 ⊢ Ord ∅ | |
2 | 1 | iordsmo 6187 | . 2 ⊢ Smo ( I ↾ ∅) |
3 | res0 4818 | . . 3 ⊢ ( I ↾ ∅) = ∅ | |
4 | smoeq 6180 | . . 3 ⊢ (( I ↾ ∅) = ∅ → (Smo ( I ↾ ∅) ↔ Smo ∅)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Smo ( I ↾ ∅) ↔ Smo ∅) |
6 | 2, 5 | mpbi 144 | 1 ⊢ Smo ∅ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∅c0 3358 I cid 4205 ↾ cres 4536 Smo wsmo 6175 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-smo 6176 |
This theorem is referenced by: (None) |
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