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Theorem nchoice 6308
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.)
Assertion
Ref Expression
nchoice <_c We NC

Proof of Theorem nchoice
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef 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
41, 2, 3sylanbrc 645 . . 3 Nn Tc 1c Tc 2c
54nrex 2716 . 2 Nn Tc 1c Tc 2c
6 nchoicelem19 6307 . . 3 <_c We NC NC Spac Fin Tc
7 finnc 6263 . . . . . . . 8 Spac Fin Nc Spac Nn
87biimpi 186 . . . . . . 7 Spac Fin Nc Spac Nn
98ad2antrl 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
1410, 11, 12, 13syl3anc 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
1514simprd 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
1716fveq2d 5372 . . . . . . . . . 10 <_c We NC NC Spac Fin Tc Spac Tc Spac
1817nceqd 6130 . . . . . . . . 9 <_c We NC NC Spac Fin Tc Nc Spac Tc Nc Spac
1918eqeq1d 2361 . . . . . . . 8 <_c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Nc Spac 1c
2018eqeq1d 2361 . . . . . . . 8 <_c We NC NC Spac Fin Tc Nc Spac Tc Tc Nc Spac 2c Nc Spac Tc Nc Spac 2c
2119, 20orbi12d 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
2215, 21mpbid 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 6179 . . . . . . . . . 10 Nc Spac Tc Tc Nc Spac
2524addceq1d 4389 . . . . . . . . 9 Nc Spac Tc 1c Tc Nc Spac 1c
2623, 25eqeq12d 2367 . . . . . . . 8 Nc Spac Tc 1c Nc Spac Tc Nc Spac 1c
2724addceq1d 4389 . . . . . . . . 9 Nc Spac Tc 2c Tc Nc Spac 2c
2823, 27eqeq12d 2367 . . . . . . . 8 Nc Spac Tc 2c Nc Spac Tc Nc Spac 2c
2926, 28orbi12d 690 . . . . . . 7 Nc Spac Tc 1c Tc 2c Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c
3029rspcev 2955 . . . . . 6 Nc Spac Nn Nc Spac Tc Nc Spac 1c Nc Spac Tc Nc Spac 2c Nn Tc 1c Tc 2c
319, 22, 30syl2anc 642 . . . . 5 <_c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c
3231ex 423 . . . 4 <_c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c
3332rexlimdva 2738 . . 3 <_c We NC NC Spac Fin Tc Nn Tc 1c Tc 2c
346, 33mpd 14 . 2 <_c We NC Nn Tc 1c Tc 2c
355, 34mto 167 1 <_c We NC
Colors of variables: wff set 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 4631  cfv 4774   We cwe 5915   NC cncs 6108   <_c clec 6109   Nc cnc 6111   Tc ctc 6113  2cc2c 6114   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-4 2135  ax-5o 2136  ax-6o 2137  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-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-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 4439  df-ltfin 4440  df-ncfin 4441  df-tfin 4442  df-evenfin 4443  df-oddfin 4444  df-sfin 4445  df-spfin 4446  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4615  df-br 4632  df-1st 4715  df-swap 4716  df-sset 4717  df-co 4718  df-ima 4719  df-si 4720  df-id 4759  df-xp 4777  df-rel 4778  df-cnv 4779  df-rn 4780  df-dm 4781  df-res 4782  df-fun 4783  df-fn 4784  df-f 4785  df-f1 4786  df-fo 4787  df-f1o 4788  df-fv 4789  df-2nd 4791  df-ov 5566  df-oprab 5567  df-mpt 5693  df-mpt2 5694  df-txp 5787  df-fix 5789  df-cup 5790  df-disj 5791  df-addcfn 5792  df-ins2 5793  df-ins3 5794  df-image 5795  df-ins4 5796  df-si3 5797  df-funs 5798  df-fns 5799  df-pw1fn 5801  df-fullfun 5802  df-clos1 5895  df-trans 5919  df-ref 5920  df-antisym 5921  df-partial 5922  df-connex 5923  df-strict 5924  df-found 5925  df-we 5926  df-sym 5928  df-er 5929  df-ec 5967  df-qs 5971  df-map 6021  df-en 6049  df-ncs 6118  df-lec 6119  df-ltc 6120  df-nc 6121  df-tc 6123  df-2c 6124  df-3c 6125  df-ce 6126  df-tcfn 6127  df-spac 6274
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