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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 9965), 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 9998. (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 9964 | . . . 4 β’ (π΄ β 1o β (cardβπ΄) = (cardβ1o)) | |
2 | 1onn 8641 | . . . . 5 β’ 1o β Ο | |
3 | cardnn 9960 | . . . . 5 β’ (1o β Ο β (cardβ1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 β’ (cardβ1o) = 1o |
5 | 1, 4 | eqtrdi 2782 | . . 3 β’ (π΄ β 1o β (cardβπ΄) = 1o) |
6 | 4 | eqeq2i 2739 | . . . . 5 β’ ((cardβπ΄) = (cardβ1o) β (cardβπ΄) = 1o) |
7 | 6 | biimpri 227 | . . . 4 β’ ((cardβπ΄) = 1o β (cardβπ΄) = (cardβ1o)) |
8 | 1n0 8489 | . . . . . . . 8 β’ 1o β β | |
9 | 8 | neii 2936 | . . . . . . 7 β’ Β¬ 1o = β |
10 | eqeq1 2730 | . . . . . . 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 6920 | . . . . . 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 8479 | . . . . . 6 β’ 1o β On | |
15 | onenon 9946 | . . . . . 6 β’ (1o β On β 1o β dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 β’ 1o β dom card |
17 | carden2 9984 | . . . . 5 β’ ((π΄ β dom card β§ 1o β dom card) β ((cardβπ΄) = (cardβ1o) β π΄ β 1o)) | |
18 | 13, 16, 17 | sylancl 585 | . . . 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 6894 | . . 3 β’ (π₯ = π΄ β ((cardβπ₯) = 1o β (cardβπ΄) = 1o)) | |
22 | 13, 21 | elab3 3671 | . 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 1533 β wcel 2098 {cab 2703 β c0 4317 class class class wbr 5141 dom cdm 5669 Oncon0 6358 βcfv 6537 Οcom 7852 1oc1o 8460 β cen 8938 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 |
This theorem is referenced by: (None) |
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