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Mirrors > Home > MPE Home > Th. List > pm54.43lem | Structured version Visualization version GIF version |
Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9909), so that their π΄ β 1 means, in our notation, π΄ β {π₯ β£ (cardβπ₯) = 1o}. Here we show that this is equivalent to π΄ β 1o so that we can use the latter more convenient notation in pm54.43 9942. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | β’ (π΄ β 1o β π΄ β {π₯ β£ (cardβπ₯) = 1o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 9908 | . . . 4 β’ (π΄ β 1o β (cardβπ΄) = (cardβ1o)) | |
2 | 1onn 8587 | . . . . 5 β’ 1o β Ο | |
3 | cardnn 9904 | . . . . 5 β’ (1o β Ο β (cardβ1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 β’ (cardβ1o) = 1o |
5 | 1, 4 | eqtrdi 2789 | . . 3 β’ (π΄ β 1o β (cardβπ΄) = 1o) |
6 | 4 | eqeq2i 2746 | . . . . 5 β’ ((cardβπ΄) = (cardβ1o) β (cardβπ΄) = 1o) |
7 | 6 | biimpri 227 | . . . 4 β’ ((cardβπ΄) = 1o β (cardβπ΄) = (cardβ1o)) |
8 | 1n0 8435 | . . . . . . . 8 β’ 1o β β | |
9 | 8 | neii 2942 | . . . . . . 7 β’ Β¬ 1o = β |
10 | eqeq1 2737 | . . . . . . 7 β’ ((cardβπ΄) = 1o β ((cardβπ΄) = β β 1o = β )) | |
11 | 9, 10 | mtbiri 327 | . . . . . 6 β’ ((cardβπ΄) = 1o β Β¬ (cardβπ΄) = β ) |
12 | ndmfv 6878 | . . . . . 6 β’ (Β¬ π΄ β dom card β (cardβπ΄) = β ) | |
13 | 11, 12 | nsyl2 141 | . . . . 5 β’ ((cardβπ΄) = 1o β π΄ β dom card) |
14 | 1on 8425 | . . . . . 6 β’ 1o β On | |
15 | onenon 9890 | . . . . . 6 β’ (1o β On β 1o β dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 β’ 1o β dom card |
17 | carden2 9928 | . . . . 5 β’ ((π΄ β dom card β§ 1o β dom card) β ((cardβπ΄) = (cardβ1o) β π΄ β 1o)) | |
18 | 13, 16, 17 | sylancl 587 | . . . 4 β’ ((cardβπ΄) = 1o β ((cardβπ΄) = (cardβ1o) β π΄ β 1o)) |
19 | 7, 18 | mpbid 231 | . . 3 β’ ((cardβπ΄) = 1o β π΄ β 1o) |
20 | 5, 19 | impbii 208 | . 2 β’ (π΄ β 1o β (cardβπ΄) = 1o) |
21 | fveqeq2 6852 | . . 3 β’ (π₯ = π΄ β ((cardβπ₯) = 1o β (cardβπ΄) = 1o)) | |
22 | 13, 21 | elab3 3639 | . 2 β’ (π΄ β {π₯ β£ (cardβπ₯) = 1o} β (cardβπ΄) = 1o) |
23 | 20, 22 | bitr4i 278 | 1 β’ (π΄ β 1o β π΄ β {π₯ β£ (cardβπ₯) = 1o}) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1542 β wcel 2107 {cab 2710 β c0 4283 class class class wbr 5106 dom cdm 5634 Oncon0 6318 βcfv 6497 Οcom 7803 1oc1o 8406 β cen 8883 cardccrd 9876 |
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-pow 5321 ax-pr 5385 ax-un 7673 |
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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 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-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 |
This theorem is referenced by: (None) |
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