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Mirrors > Home > NFE Home > Th. List > nchoice | Unicode version |
Description: Cardinal less than or equal does not well-order the cardinals. This is equivalent to saying that the axiom of choice from ZFC is false in NF. Theorem 7.5 of [Specker] p. 974. (Contributed by SF, 20-Mar-2015.) |
Ref | Expression |
---|---|
nchoice | c We NC |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nchoicelem1 6289 | . . . 4 Nn Tc 1c | |
2 | nchoicelem2 6290 | . . . 4 Nn Tc 2c | |
3 | ioran 476 | . . . 4 Tc 1c Tc 2c Tc 1c Tc 2c | |
4 | 1, 2, 3 | sylanbrc 645 | . . 3 Nn Tc 1c Tc 2c |
5 | 4 | nrex 2716 | . 2 Nn Tc 1c Tc 2c |
6 | nchoicelem19 6307 | . . 3 c We NC NC Spac Fin Tc | |
7 | finnc 6243 | . . . . . . . 8 Spac Fin Nc Spac Nn | |
8 | 7 | biimpi 186 | . . . . . . 7 Spac Fin Nc Spac Nn |
9 | 8 | ad2antrl 708 | . . . . . 6 c We NC NC Spac Fin Tc Nc Spac Nn |
10 | simpll 730 | . . . . . . . . 9 c We NC NC Spac Fin Tc c We NC | |
11 | simplr 731 | . . . . . . . . 9 c We NC NC Spac Fin Tc NC | |
12 | simprl 732 | . . . . . . . . 9 c We NC NC Spac Fin Tc Spac Fin | |
13 | nchoicelem17 6305 | . . . . . . . . 9 c We NC NC Spac Fin Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c | |
14 | 10, 11, 12, 13 | syl3anc 1182 | . . . . . . . 8 c We NC NC Spac Fin Tc Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c |
15 | 14 | simprd 449 | . . . . . . 7 c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c |
16 | simprr 733 | . . . . . . . . . . 11 c We NC NC Spac Fin Tc Tc | |
17 | 16 | fveq2d 5332 | . . . . . . . . . 10 c We NC NC Spac Fin Tc Spac Tc Spac |
18 | 17 | nceqd 6110 | . . . . . . . . 9 c We NC NC Spac Fin Tc Nc Spac Tc Nc Spac |
19 | 18 | eqeq1d 2361 | . . . . . . . 8 c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Nc Spac 1c |
20 | 18 | eqeq1d 2361 | . . . . . . . 8 c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 2c Nc Spac Tc Nc Spac 2c |
21 | 19, 20 | orbi12d 690 | . . . . . . 7 c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c |
22 | 15, 21 | mpbid 201 | . . . . . 6 c We NC NC Spac Fin Tc Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c |
23 | id 19 | . . . . . . . . 9 Nc Spac Nc Spac | |
24 | tceq 6158 | . . . . . . . . . 10 Nc Spac Tc Tc Nc Spac | |
25 | 24 | addceq1d 4389 | . . . . . . . . 9 Nc Spac Tc 1c Tc Nc Spac 1c |
26 | 23, 25 | eqeq12d 2367 | . . . . . . . 8 Nc Spac Tc 1c Nc Spac Tc Nc Spac 1c |
27 | 24 | addceq1d 4389 | . . . . . . . . 9 Nc Spac Tc 2c Tc Nc Spac 2c |
28 | 23, 27 | eqeq12d 2367 | . . . . . . . 8 Nc Spac Tc 2c Nc Spac Tc Nc Spac 2c |
29 | 26, 28 | orbi12d 690 | . . . . . . 7 Nc Spac Tc 1c Tc 2c Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c |
30 | 29 | rspcev 2955 | . . . . . 6 Nc Spac Nn Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c Nn Tc 1c Tc 2c |
31 | 9, 22, 30 | syl2anc 642 | . . . . 5 c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c |
32 | 31 | ex 423 | . . . 4 c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c |
33 | 32 | rexlimdva 2738 | . . 3 c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c |
34 | 6, 33 | mpd 14 | . 2 c We NC Nn Tc 1c Tc 2c |
35 | 5, 34 | mto 167 | 1 c We NC |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wo 357 wa 358 wceq 1642 wcel 1710 wrex 2615 1cc1c 4134 Nn cnnc 4373 cplc 4375 Fin cfin 4376 class class class wbr 4639 cfv 4781 We cwe 5895 NC cncs 6088 c clec 6089 Nc cnc 6091 Tc ctc 6093 2cc2c 6094 Spac cspac 6273 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-meredith 1406 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-tp 3743 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-fix 5740 df-cup 5742 df-disj 5744 df-addcfn 5746 df-compose 5748 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-pw1fn 5766 df-fullfun 5768 df-clos1 5873 df-trans 5899 df-ref 5900 df-antisym 5901 df-partial 5902 df-connex 5903 df-strict 5904 df-found 5905 df-we 5906 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-map 6001 df-en 6029 df-ncs 6098 df-lec 6099 df-ltc 6100 df-nc 6101 df-tc 6103 df-2c 6104 df-3c 6105 df-ce 6106 df-tcfn 6107 df-spac 6274 |
This theorem is referenced by: (None) |
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