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Mirrors > Home > MPE Home > Th. List > smofvon2 | Structured version Visualization version GIF version |
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Ref | Expression |
---|---|
smofvon2 | β’ (Smo πΉ β (πΉβπ΅) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsmo2 8294 | . . . 4 β’ (Smo πΉ β (πΉ:dom πΉβΆOn β§ Ord dom πΉ β§ βπ₯ β dom πΉβπ¦ β π₯ (πΉβπ¦) β (πΉβπ₯))) | |
2 | 1 | simp1bi 1146 | . . 3 β’ (Smo πΉ β πΉ:dom πΉβΆOn) |
3 | ffvelcdm 7033 | . . . 4 β’ ((πΉ:dom πΉβΆOn β§ π΅ β dom πΉ) β (πΉβπ΅) β On) | |
4 | 3 | expcom 415 | . . 3 β’ (π΅ β dom πΉ β (πΉ:dom πΉβΆOn β (πΉβπ΅) β On)) |
5 | 2, 4 | syl5 34 | . 2 β’ (π΅ β dom πΉ β (Smo πΉ β (πΉβπ΅) β On)) |
6 | ndmfv 6878 | . . . 4 β’ (Β¬ π΅ β dom πΉ β (πΉβπ΅) = β ) | |
7 | 0elon 6372 | . . . 4 β’ β β On | |
8 | 6, 7 | eqeltrdi 2842 | . . 3 β’ (Β¬ π΅ β dom πΉ β (πΉβπ΅) β On) |
9 | 8 | a1d 25 | . 2 β’ (Β¬ π΅ β dom πΉ β (Smo πΉ β (πΉβπ΅) β On)) |
10 | 5, 9 | pm2.61i 182 | 1 β’ (Smo πΉ β (πΉβπ΅) β On) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wcel 2107 βwral 3061 β c0 4283 dom cdm 5634 Ord word 6317 Oncon0 6318 βΆwf 6493 βcfv 6497 Smo wsmo 8292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-smo 8293 |
This theorem is referenced by: smo11 8311 smoord 8312 smoword 8313 smogt 8314 |
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