Theorem nchoicelem8 6297

Description: Lemma for nchoice 6309. An anti-closure condition for cardinal exponentiation to zero. Theorem 4.5 of [Specker] p. 973. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
nchoicelem8	⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ (M ↑c 0c) ∈ NC ↔ Nc 1c <c M))

Proof of Theorem nchoicelem8

Step	Hyp	Ref	Expression
1		ce0lenc1 6240	. . . 4 ⊢ (M ∈ NC → ((M ↑c 0c) ∈ NC ↔ M ≤c Nc 1c))
2	1	notbid 285	. . 3 ⊢ (M ∈ NC → (¬ (M ↑c 0c) ∈ NC ↔ ¬ M ≤c Nc 1c))
3	2	adantl 452	. 2 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ (M ↑c 0c) ∈ NC ↔ ¬ M ≤c Nc 1c))
4		df-we 5907	. . . . . . . . . 10 ⊢ We = ( Or ∩ Fr )
5	4	breqi 4646	. . . . . . . . 9 ⊢ ( ≤c We NC ↔ ≤c ( Or ∩ Fr ) NC )
6		brin 4694	. . . . . . . . 9 ⊢ ( ≤c ( Or ∩ Fr ) NC ↔ ( ≤c Or NC ∧ ≤c Fr NC ))
7	5, 6	bitri 240	. . . . . . . 8 ⊢ ( ≤c We NC ↔ ( ≤c Or NC ∧ ≤c Fr NC ))
8		sopc 5935	. . . . . . . . . 10 ⊢ ( ≤c Or NC ↔ ( ≤c Po NC ∧ ≤c Connex NC ))
9	8	simprbi 450	. . . . . . . . 9 ⊢ ( ≤c Or NC → ≤c Connex NC )
10	9	adantr 451	. . . . . . . 8 ⊢ (( ≤c Or NC ∧ ≤c Fr NC ) → ≤c Connex NC )
11	7, 10	sylbi 187	. . . . . . 7 ⊢ ( ≤c We NC → ≤c Connex NC )
12		simpl 443	. . . . . . . 8 ⊢ (( ≤c Connex NC ∧ M ∈ NC ) → ≤c Connex NC )
13		simpr 447	. . . . . . . 8 ⊢ (( ≤c Connex NC ∧ M ∈ NC ) → M ∈ NC )
14		1cex 4143	. . . . . . . . . 10 ⊢ 1c ∈ V
15	14	ncelncsi 6122	. . . . . . . . 9 ⊢ Nc 1c ∈ NC
16	15	a1i 10	. . . . . . . 8 ⊢ (( ≤c Connex NC ∧ M ∈ NC ) → Nc 1c ∈ NC )
17	12, 13, 16	connexd 5932	. . . . . . 7 ⊢ (( ≤c Connex NC ∧ M ∈ NC ) → (M ≤c Nc 1c ∨ Nc 1c ≤c M))
18	11, 17	sylan 457	. . . . . 6 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (M ≤c Nc 1c ∨ Nc 1c ≤c M))
19	18	ord 366	. . . . 5 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ M ≤c Nc 1c → Nc 1c ≤c M))
20		id 19	. . . . . . . 8 ⊢ ( Nc 1c = M → Nc 1c = M)
21		nclecid 6198	. . . . . . . . 9 ⊢ ( Nc 1c ∈ NC → Nc 1c ≤c Nc 1c)
22	15, 21	ax-mp 5	. . . . . . . 8 ⊢ Nc 1c ≤c Nc 1c
23	20, 22	syl6eqbrr 4678	. . . . . . 7 ⊢ ( Nc 1c = M → M ≤c Nc 1c)
24	23	necon3bi 2558	. . . . . 6 ⊢ (¬ M ≤c Nc 1c → Nc 1c ≠ M)
25	24	a1i 10	. . . . 5 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ M ≤c Nc 1c → Nc 1c ≠ M))
26	19, 25	jcad 519	. . . 4 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ M ≤c Nc 1c → ( Nc 1c ≤c M ∧ Nc 1c ≠ M)))
27	7	simplbi 446	. . . . . . . . . . 11 ⊢ ( ≤c We NC → ≤c Or NC )
28	8	simplbi 446	. . . . . . . . . . 11 ⊢ ( ≤c Or NC → ≤c Po NC )
29		df-partial 5903	. . . . . . . . . . . . . 14 ⊢ Po = (( Ref ∩ Trans ) ∩ Antisym )
30	29	breqi 4646	. . . . . . . . . . . . 13 ⊢ ( ≤c Po NC ↔ ≤c (( Ref ∩ Trans ) ∩ Antisym ) NC )
31		brin 4694	. . . . . . . . . . . . 13 ⊢ ( ≤c (( Ref ∩ Trans ) ∩ Antisym ) NC ↔ ( ≤c ( Ref ∩ Trans ) NC ∧ ≤c Antisym NC ))
32	30, 31	bitri 240	. . . . . . . . . . . 12 ⊢ ( ≤c Po NC ↔ ( ≤c ( Ref ∩ Trans ) NC ∧ ≤c Antisym NC ))
33	32	simprbi 450	. . . . . . . . . . 11 ⊢ ( ≤c Po NC → ≤c Antisym NC )
34	27, 28, 33	3syl 18	. . . . . . . . . 10 ⊢ ( ≤c We NC → ≤c Antisym NC )
35	34	adantr 451	. . . . . . . . 9 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ≤c Antisym NC )
36	35	adantr 451	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → ≤c Antisym NC )
37	15	a1i 10	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → Nc 1c ∈ NC )
38		simplr 731	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → M ∈ NC )
39		simprl 732	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → Nc 1c ≤c M)
40		simprr 733	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → M ≤c Nc 1c)
41	36, 37, 38, 39, 40	antid 5930	. . . . . . 7 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ ( Nc 1c ≤c M ∧ M ≤c Nc 1c)) → Nc 1c = M)
42	41	expr 598	. . . . . 6 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ Nc 1c ≤c M) → (M ≤c Nc 1c → Nc 1c = M))
43	42	necon3ad 2553	. . . . 5 ⊢ ((( ≤c We NC ∧ M ∈ NC ) ∧ Nc 1c ≤c M) → ( Nc 1c ≠ M → ¬ M ≤c Nc 1c))
44	43	expimpd 586	. . . 4 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (( Nc 1c ≤c M ∧ Nc 1c ≠ M) → ¬ M ≤c Nc 1c))
45	26, 44	impbid 183	. . 3 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ M ≤c Nc 1c ↔ ( Nc 1c ≤c M ∧ Nc 1c ≠ M)))
46		brltc 6115	. . 3 ⊢ ( Nc 1c <c M ↔ ( Nc 1c ≤c M ∧ Nc 1c ≠ M))
47	45, 46	syl6bbr 254	. 2 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ M ≤c Nc 1c ↔ Nc 1c <c M))
48	3, 47	bitrd 244	1 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ (M ↑c 0c) ∈ NC ↔ Nc 1c <c M))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ∩ cin 3209  1_cc1c 4135  0_cc0c 4375   class class class wbr 4640  (class class class)co 5526   Trans ctrans 5889   Ref cref 5890   Antisym cantisym 5891   Po cpartial 5892   Connex cconnex 5893   Or cstrict 5894   Fr cfound 5895   We cwe 5896   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091   Nc cnc 6092   ↑_c cce 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-antisym 5902  df-partial 5903  df-connex 5904  df-strict 5905  df-we 5907  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-ce 6107

This theorem is referenced by:  nchoicelem9  6298  nchoicelem19  6308