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Mirrors > Home > MPE Home > Th. List > harword | Structured version Visualization version GIF version |
Description: Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
harword | β’ (π βΌ π β (harβπ) β (harβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domtr 8953 | . . . . 5 β’ ((π¦ βΌ π β§ π βΌ π) β π¦ βΌ π) | |
2 | 1 | expcom 415 | . . . 4 β’ (π βΌ π β (π¦ βΌ π β π¦ βΌ π)) |
3 | 2 | adantr 482 | . . 3 β’ ((π βΌ π β§ π¦ β On) β (π¦ βΌ π β π¦ βΌ π)) |
4 | 3 | ss2rabdv 4037 | . 2 β’ (π βΌ π β {π¦ β On β£ π¦ βΌ π} β {π¦ β On β£ π¦ βΌ π}) |
5 | reldom 8895 | . . . 4 β’ Rel βΌ | |
6 | 5 | brrelex1i 5692 | . . 3 β’ (π βΌ π β π β V) |
7 | harval 9504 | . . 3 β’ (π β V β (harβπ) = {π¦ β On β£ π¦ βΌ π}) | |
8 | 6, 7 | syl 17 | . 2 β’ (π βΌ π β (harβπ) = {π¦ β On β£ π¦ βΌ π}) |
9 | 5 | brrelex2i 5693 | . . 3 β’ (π βΌ π β π β V) |
10 | harval 9504 | . . 3 β’ (π β V β (harβπ) = {π¦ β On β£ π¦ βΌ π}) | |
11 | 9, 10 | syl 17 | . 2 β’ (π βΌ π β (harβπ) = {π¦ β On β£ π¦ βΌ π}) |
12 | 4, 8, 11 | 3sstr4d 3995 | 1 β’ (π βΌ π β (harβπ) β (harβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3447 β wss 3914 class class class wbr 5109 Oncon0 6321 βcfv 6500 βΌ cdom 8887 harchar 9500 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-en 8890 df-dom 8891 df-oi 9454 df-har 9501 |
This theorem is referenced by: hsmexlem3 10372 |
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