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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 m n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nchoicelem1 6289 . . . 4 (n Nn → ¬ n = ( Tc n +c 1c))
2 nchoicelem2 6290 . . . 4 (n Nn → ¬ n = ( Tc n +c 2c))
3 ioran 476 . . . 4 (¬ (n = ( Tc n +c 1c) n = ( Tc n +c 2c)) ↔ (¬ n = ( Tc n +c 1c) ¬ n = ( Tc n +c 2c)))
41, 2, 3sylanbrc 645 . . 3 (n Nn → ¬ (n = ( Tc n +c 1c) n = ( Tc n +c 2c)))
54nrex 2716 . 2 ¬ n Nn (n = ( Tc n +c 1c) n = ( Tc n +c 2c))
6 nchoicelem19 6307 . . 3 ( ≤c We NCm NC (( Spacm) Fin Tc m = m))
7 finnc 6263 . . . . . . . 8 (( Spacm) FinNc ( Spacm) Nn )
87biimpi 186 . . . . . . 7 (( Spacm) FinNc ( Spacm) Nn )
98ad2antrl 708 . . . . . 6 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → Nc ( Spacm) Nn )
10 simpll 730 . . . . . . . . 9 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ≤c We NC )
11 simplr 731 . . . . . . . . 9 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → m NC )
12 simprl 732 . . . . . . . . 9 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( Spacm) Fin )
13 nchoicelem17 6305 . . . . . . . . 9 (( ≤c We NC m NC ( Spacm) Fin ) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))
1410, 11, 12, 13syl3anc 1182 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → (( SpacTc m) Fin ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c))))
1514simprd 449 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)))
16 simprr 733 . . . . . . . . . . 11 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → Tc m = m)
1716fveq2d 5372 . . . . . . . . . 10 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( SpacTc m) = ( Spacm))
1817nceqd 6130 . . . . . . . . 9 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → Nc ( SpacTc m) = Nc ( Spacm))
1918eqeq1d 2361 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) ↔ Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c)))
2018eqeq1d 2361 . . . . . . . 8 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c) ↔ Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c)))
2119, 20orbi12d 690 . . . . . . 7 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → (( Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 1c) Nc ( SpacTc m) = ( Tc Nc ( Spacm) +c 2c)) ↔ ( Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c) Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c))))
2215, 21mpbid 201 . . . . . 6 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → ( Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c) Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c)))
23 id 19 . . . . . . . . 9 (n = Nc ( Spacm) → n = Nc ( Spacm))
24 tceq 6179 . . . . . . . . . 10 (n = Nc ( Spacm) → Tc n = Tc Nc ( Spacm))
2524addceq1d 4389 . . . . . . . . 9 (n = Nc ( Spacm) → ( Tc n +c 1c) = ( Tc Nc ( Spacm) +c 1c))
2623, 25eqeq12d 2367 . . . . . . . 8 (n = Nc ( Spacm) → (n = ( Tc n +c 1c) ↔ Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c)))
2724addceq1d 4389 . . . . . . . . 9 (n = Nc ( Spacm) → ( Tc n +c 2c) = ( Tc Nc ( Spacm) +c 2c))
2823, 27eqeq12d 2367 . . . . . . . 8 (n = Nc ( Spacm) → (n = ( Tc n +c 2c) ↔ Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c)))
2926, 28orbi12d 690 . . . . . . 7 (n = Nc ( Spacm) → ((n = ( Tc n +c 1c) n = ( Tc n +c 2c)) ↔ ( Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c) Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c))))
3029rspcev 2955 . . . . . 6 (( Nc ( Spacm) Nn ( Nc ( Spacm) = ( Tc Nc ( Spacm) +c 1c) Nc ( Spacm) = ( Tc Nc ( Spacm) +c 2c))) → n Nn (n = ( Tc n +c 1c) n = ( Tc n +c 2c)))
319, 22, 30syl2anc 642 . . . . 5 ((( ≤c We NC m NC ) (( Spacm) Fin Tc m = m)) → n Nn (n = ( Tc n +c 1c) n = ( Tc n +c 2c)))
3231ex 423 . . . 4 (( ≤c We NC m NC ) → ((( Spacm) Fin Tc m = m) → n Nn (n = ( Tc n +c 1c) n = ( Tc n +c 2c))))
3332rexlimdva 2738 . . 3 ( ≤c We NC → (m NC (( Spacm) Fin Tc m = m) → n Nn (n = ( Tc n +c 1c) n = ( Tc n +c 2c))))
346, 33mpd 14 . 2 ( ≤c We NCn Nn (n = ( Tc n +c 1c) n = ( Tc n +c 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   +c 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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