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Mirrors > Home > MPE Home > Th. List > issmo | Structured version Visualization version GIF version |
Description: Conditions for which π΄ is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2371. (Revised by Gino Giotto, 19-May-2023.) |
Ref | Expression |
---|---|
issmo.1 | β’ π΄:π΅βΆOn |
issmo.2 | β’ Ord π΅ |
issmo.3 | β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β π¦ β (π΄βπ₯) β (π΄βπ¦))) |
issmo.4 | β’ dom π΄ = π΅ |
Ref | Expression |
---|---|
issmo | β’ Smo π΄ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmo.1 | . . 3 β’ π΄:π΅βΆOn | |
2 | issmo.4 | . . . 4 β’ dom π΄ = π΅ | |
3 | 2 | feq2i 6709 | . . 3 β’ (π΄:dom π΄βΆOn β π΄:π΅βΆOn) |
4 | 1, 3 | mpbir 230 | . 2 β’ π΄:dom π΄βΆOn |
5 | issmo.2 | . . 3 β’ Ord π΅ | |
6 | ordeq 6371 | . . . 4 β’ (dom π΄ = π΅ β (Ord dom π΄ β Ord π΅)) | |
7 | 2, 6 | ax-mp 5 | . . 3 β’ (Ord dom π΄ β Ord π΅) |
8 | 5, 7 | mpbir 230 | . 2 β’ Ord dom π΄ |
9 | 2 | eleq2i 2825 | . . . 4 β’ (π₯ β dom π΄ β π₯ β π΅) |
10 | 2 | eleq2i 2825 | . . . 4 β’ (π¦ β dom π΄ β π¦ β π΅) |
11 | issmo.3 | . . . 4 β’ ((π₯ β π΅ β§ π¦ β π΅) β (π₯ β π¦ β (π΄βπ₯) β (π΄βπ¦))) | |
12 | 9, 10, 11 | syl2anb 598 | . . 3 β’ ((π₯ β dom π΄ β§ π¦ β dom π΄) β (π₯ β π¦ β (π΄βπ₯) β (π΄βπ¦))) |
13 | 12 | rgen2 3197 | . 2 β’ βπ₯ β dom π΄βπ¦ β dom π΄(π₯ β π¦ β (π΄βπ₯) β (π΄βπ¦)) |
14 | df-smo 8345 | . 2 β’ (Smo π΄ β (π΄:dom π΄βΆOn β§ Ord dom π΄ β§ βπ₯ β dom π΄βπ¦ β dom π΄(π₯ β π¦ β (π΄βπ₯) β (π΄βπ¦)))) | |
15 | 4, 8, 13, 14 | mpbir3an 1341 | 1 β’ Smo π΄ |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 dom cdm 5676 Ord word 6363 Oncon0 6364 βΆwf 6539 βcfv 6543 Smo wsmo 8344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-tr 5266 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-fn 6546 df-f 6547 df-smo 8345 |
This theorem is referenced by: iordsmo 8356 smobeth 10580 |
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