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Mirrors > Home > MPE Home > Th. List > card2inf | Unicode version |
Description: The definition cardval2 8393 has the curious property that for non-numerable sets (for which ndmfv 5895 yields ), it still evaluates to a nonempty set, and indeed it contains . (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
card2inf.1 |
Ref | Expression |
---|---|
card2inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4455 | . . . . 5 | |
2 | breq1 4455 | . . . . 5 | |
3 | breq1 4455 | . . . . 5 | |
4 | 0elon 4936 | . . . . . . . 8 | |
5 | breq1 4455 | . . . . . . . . 9 | |
6 | 5 | rspcev 3210 | . . . . . . . 8 |
7 | 4, 6 | mpan 670 | . . . . . . 7 |
8 | 7 | con3i 135 | . . . . . 6 |
9 | card2inf.1 | . . . . . . . 8 | |
10 | 9 | 0dom 7667 | . . . . . . 7 |
11 | brsdom 7558 | . . . . . . 7 | |
12 | 10, 11 | mpbiran 918 | . . . . . 6 |
13 | 8, 12 | sylibr 212 | . . . . 5 |
14 | sucdom2 7734 | . . . . . . . 8 | |
15 | 14 | ad2antll 728 | . . . . . . 7 |
16 | nnon 6706 | . . . . . . . . . 10 | |
17 | suceloni 6648 | . . . . . . . . . 10 | |
18 | breq1 4455 | . . . . . . . . . . . 12 | |
19 | 18 | rspcev 3210 | . . . . . . . . . . 11 |
20 | 19 | ex 434 | . . . . . . . . . 10 |
21 | 16, 17, 20 | 3syl 20 | . . . . . . . . 9 |
22 | 21 | con3dimp 441 | . . . . . . . 8 |
23 | 22 | adantrr 716 | . . . . . . 7 |
24 | brsdom 7558 | . . . . . . 7 | |
25 | 15, 23, 24 | sylanbrc 664 | . . . . . 6 |
26 | 25 | exp32 605 | . . . . 5 |
27 | 1, 2, 3, 13, 26 | finds2 6728 | . . . 4 |
28 | 27 | com12 31 | . . 3 |
29 | 28 | ralrimiv 2869 | . 2 |
30 | omsson 6704 | . . 3 | |
31 | ssrab 3577 | . . 3 | |
32 | 30, 31 | mpbiran 918 | . 2 |
33 | 29, 32 | sylibr 212 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
/\ wa 369 e. wcel 1818 A. wral 2807
E. wrex 2808 { crab 2811 cvv 3109
C_ wss 3475 c0 3784 class class class wbr 4452
con0 4883 suc csuc 4885 com 6700
cen 7533 cdom 7534 csdm 7535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-sbc 3328 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-om 6701 df-1o 7149 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 |
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