Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > issmo | Structured version Visualization version Unicode version |
Description: Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Ref | Expression |
---|---|
issmo.1 | |
issmo.2 | |
issmo.3 | |
issmo.4 |
Ref | Expression |
---|---|
issmo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmo.1 | . . 3 | |
2 | issmo.4 | . . . 4 | |
3 | 2 | feq2i 6037 | . . 3 |
4 | 1, 3 | mpbir 221 | . 2 |
5 | issmo.2 | . . 3 | |
6 | ordeq 5730 | . . . 4 | |
7 | 2, 6 | ax-mp 5 | . . 3 |
8 | 5, 7 | mpbir 221 | . 2 |
9 | 2 | eleq2i 2693 | . . . 4 |
10 | 2 | eleq2i 2693 | . . . 4 |
11 | issmo.3 | . . . 4 | |
12 | 9, 10, 11 | syl2anb 496 | . . 3 |
13 | 12 | rgen2a 2977 | . 2 |
14 | df-smo 7443 | . 2 | |
15 | 4, 8, 13, 14 | mpbir3an 1244 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cdm 5114 word 5722 con0 5723 wf 5884 cfv 5888 wsmo 7442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-in 3581 df-ss 3588 df-uni 4437 df-tr 4753 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-fn 5891 df-f 5892 df-smo 7443 |
This theorem is referenced by: iordsmo 7454 smobeth 9408 |
Copyright terms: Public domain | W3C validator |