![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > smo0 | Structured version Visualization version 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 6015 | . . 3 ⊢ Ord ∅ | |
2 | 1 | iordsmo 7720 | . 2 ⊢ Smo ( I ↾ ∅) |
3 | res0 5633 | . . 3 ⊢ ( I ↾ ∅) = ∅ | |
4 | smoeq 7713 | . . 3 ⊢ (( I ↾ ∅) = ∅ → (Smo ( I ↾ ∅) ↔ Smo ∅)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Smo ( I ↾ ∅) ↔ Smo ∅) |
6 | 2, 5 | mpbi 222 | 1 ⊢ Smo ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∅c0 4144 I cid 5249 ↾ cres 5344 Smo wsmo 7708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-smo 7709 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |