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 6236 | . . 3 ⊢ Ord ∅ | |
2 | 1 | iordsmo 7983 | . 2 ⊢ Smo ( I ↾ ∅) |
3 | res0 5850 | . . 3 ⊢ ( I ↾ ∅) = ∅ | |
4 | smoeq 7976 | . . 3 ⊢ (( I ↾ ∅) = ∅ → (Smo ( I ↾ ∅) ↔ Smo ∅)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Smo ( I ↾ ∅) ↔ Smo ∅) |
6 | 2, 5 | mpbi 231 | 1 ⊢ Smo ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∅c0 4288 I cid 5452 ↾ cres 5550 Smo wsmo 7971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-smo 7972 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |