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.

Step	Hyp	Ref	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)))
4	1, 2, 3	sylanbrc 645	. . 3 ⊢ (n ∈ Nn → ¬ (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c)))
5	4	nrex 2716	. 2 ⊢ ¬ ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c))
6		nchoicelem19 6307	. . 3 ⊢ ( ≤c We NC → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m = m))
7		finnc 6243	. . . . . . . 8 ⊢ (( Spac ‘m) ∈ Fin ↔ Nc ( Spac ‘m) ∈ Nn )
8	7	biimpi 186	. . . . . . 7 ⊢ (( Spac ‘m) ∈ Fin → Nc ( Spac ‘m) ∈ Nn )
9	8	ad2antrl 708	. . . . . 6 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → Nc ( Spac ‘m) ∈ Nn )
10		simpll 730	. . . . . . . . 9 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ≤c We NC )
11		simplr 731	. . . . . . . . 9 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → m ∈ NC )
12		simprl 732	. . . . . . . . 9 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Spac ‘m) ∈ Fin )
13		nchoicelem17 6305	. . . . . . . . 9 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ( Spac ‘m) ∈ Fin ) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))
14	10, 11, 12, 13	syl3anc 1182	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))
15	14	simprd 449	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)))
16		simprr 733	. . . . . . . . . . 11 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → Tc m = m)
17	16	fveq2d 5332	. . . . . . . . . 10 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Spac ‘ Tc m) = ( Spac ‘m))
18	17	nceqd 6110	. . . . . . . . 9 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘m))
19	18	eqeq1d 2361	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ↔ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c)))
20	18	eqeq1d 2361	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c) ↔ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c)))
21	19, 20	orbi12d 690	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → (( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c)) ↔ ( Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c))))
22	15, 21	mpbid 201	. . . . . 6 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ( Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c)))
23		id 19	. . . . . . . . 9 ⊢ (n = Nc ( Spac ‘m) → n = Nc ( Spac ‘m))
24		tceq 6158	. . . . . . . . . 10 ⊢ (n = Nc ( Spac ‘m) → Tc n = Tc Nc ( Spac ‘m))
25	24	addceq1d 4389	. . . . . . . . 9 ⊢ (n = Nc ( Spac ‘m) → ( Tc n +c 1c) = ( Tc Nc ( Spac ‘m) +c 1c))
26	23, 25	eqeq12d 2367	. . . . . . . 8 ⊢ (n = Nc ( Spac ‘m) → (n = ( Tc n +c 1c) ↔ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c)))
27	24	addceq1d 4389	. . . . . . . . 9 ⊢ (n = Nc ( Spac ‘m) → ( Tc n +c 2c) = ( Tc Nc ( Spac ‘m) +c 2c))
28	23, 27	eqeq12d 2367	. . . . . . . 8 ⊢ (n = Nc ( Spac ‘m) → (n = ( Tc n +c 2c) ↔ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c)))
29	26, 28	orbi12d 690	. . . . . . 7 ⊢ (n = Nc ( Spac ‘m) → ((n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c)) ↔ ( Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c))))
30	29	rspcev 2955	. . . . . 6 ⊢ (( Nc ( Spac ‘m) ∈ Nn ∧ ( Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘m) = ( Tc Nc ( Spac ‘m) +c 2c))) → ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c)))
31	9, 22, 30	syl2anc 642	. . . . 5 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ Tc m = m)) → ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c)))
32	31	ex 423	. . . 4 ⊢ (( ≤c We NC ∧ m ∈ NC ) → ((( Spac ‘m) ∈ Fin ∧ Tc m = m) → ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c))))
33	32	rexlimdva 2738	. . 3 ⊢ ( ≤c We NC → (∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m = m) → ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c))))
34	6, 33	mpd 14	. 2 ⊢ ( ≤c We NC → ∃n ∈ Nn (n = ( Tc n +c 1c) ∨ n = ( Tc n +c 2c)))
35	5, 34	mto 167	1 ⊢ ¬ ≤c We NC

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  1_cc1c 4134   Nn cnnc 4373   +_c cplc 4375   Fin cfin 4376   class class class wbr 4639   ‘cfv 4781   We cwe 5895   NC cncs 6088   ≤_c clec 6089   Nc cnc 6091   T_c ctc 6093  2_cc2c 6094   Sp_ac cspac 6273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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)