Theorem nchoicelem3 6292

Description: Lemma for nchoice 6309. Compute the value of Sp_ac when the argument is not exponentiable. Theorem 6.2 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
nchoicelem3	⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = {M})

Proof of Theorem nchoicelem3

Dummy variables p x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		spacval 6283	. . 3 ⊢ (M ∈ NC → ( Spac ‘M) = Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}))
2	1	adantr 451	. 2 ⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}))
3		elimasn 5020	. . . . . . . 8 ⊢ (p ∈ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}) ↔ ⟨M, p⟩ ∈ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})
4		df-br 4641	. . . . . . . 8 ⊢ (M{⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}p ↔ ⟨M, p⟩ ∈ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})
5	3, 4	bitr4i 243	. . . . . . 7 ⊢ (p ∈ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}) ↔ M{⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}p)
6		vex 2863	. . . . . . . . 9 ⊢ p ∈ V
7		eleq1 2413	. . . . . . . . . . 11 ⊢ (x = M → (x ∈ NC ↔ M ∈ NC ))
8		oveq2 5532	. . . . . . . . . . . 12 ⊢ (x = M → (2c ↑c x) = (2c ↑c M))
9	8	eqeq2d 2364	. . . . . . . . . . 11 ⊢ (x = M → (y = (2c ↑c x) ↔ y = (2c ↑c M)))
10	7, 9	3anbi13d 1254	. . . . . . . . . 10 ⊢ (x = M → ((x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x)) ↔ (M ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c M))))
11		eleq1 2413	. . . . . . . . . . 11 ⊢ (y = p → (y ∈ NC ↔ p ∈ NC ))
12		eqeq1 2359	. . . . . . . . . . 11 ⊢ (y = p → (y = (2c ↑c M) ↔ p = (2c ↑c M)))
13	11, 12	3anbi23d 1255	. . . . . . . . . 10 ⊢ (y = p → ((M ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c M)) ↔ (M ∈ NC ∧ p ∈ NC ∧ p = (2c ↑c M))))
14		eqid 2353	. . . . . . . . . 10 ⊢ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} = {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}
15	10, 13, 14	brabg 4707	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ p ∈ V) → (M{⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}p ↔ (M ∈ NC ∧ p ∈ NC ∧ p = (2c ↑c M))))
16	6, 15	mpan2 652	. . . . . . . 8 ⊢ (M ∈ NC → (M{⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}p ↔ (M ∈ NC ∧ p ∈ NC ∧ p = (2c ↑c M))))
17		eleq1 2413	. . . . . . . . . . 11 ⊢ (p = (2c ↑c M) → (p ∈ NC ↔ (2c ↑c M) ∈ NC ))
18	17	biimpac 472	. . . . . . . . . 10 ⊢ ((p ∈ NC ∧ p = (2c ↑c M)) → (2c ↑c M) ∈ NC )
19		2nc 6169	. . . . . . . . . . 11 ⊢ 2c ∈ NC
20		ceclr 6188	. . . . . . . . . . . 12 ⊢ ((2c ∈ NC ∧ M ∈ NC ∧ (2c ↑c M) ∈ NC ) → ((2c ↑c 0c) ∈ NC ∧ (M ↑c 0c) ∈ NC ))
21	20	simprd 449	. . . . . . . . . . 11 ⊢ ((2c ∈ NC ∧ M ∈ NC ∧ (2c ↑c M) ∈ NC ) → (M ↑c 0c) ∈ NC )
22	19, 21	mp3an1 1264	. . . . . . . . . 10 ⊢ ((M ∈ NC ∧ (2c ↑c M) ∈ NC ) → (M ↑c 0c) ∈ NC )
23	18, 22	sylan2 460	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ (p ∈ NC ∧ p = (2c ↑c M))) → (M ↑c 0c) ∈ NC )
24	23	3impb 1147	. . . . . . . 8 ⊢ ((M ∈ NC ∧ p ∈ NC ∧ p = (2c ↑c M)) → (M ↑c 0c) ∈ NC )
25	16, 24	syl6bi 219	. . . . . . 7 ⊢ (M ∈ NC → (M{⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}p → (M ↑c 0c) ∈ NC ))
26	5, 25	syl5bi 208	. . . . . 6 ⊢ (M ∈ NC → (p ∈ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}) → (M ↑c 0c) ∈ NC ))
27	26	con3d 125	. . . . 5 ⊢ (M ∈ NC → (¬ (M ↑c 0c) ∈ NC → ¬ p ∈ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M})))
28	27	imp 418	. . . 4 ⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ¬ p ∈ ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}))
29	28	eq0rdv 3586	. . 3 ⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}) = ∅)
30		snex 4112	. . . 4 ⊢ {M} ∈ V
31		spacvallem1 6282	. . . 4 ⊢ {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} ∈ V
32		eqid 2353	. . . 4 ⊢ Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) = Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))})
33	30, 31, 32	clos1nrel 5887	. . 3 ⊢ (({⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))} “ {M}) = ∅ → Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) = {M})
34	29, 33	syl 15	. 2 ⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → Clos1 ({M}, {⟨x, y⟩ ∣ (x ∈ NC ∧ y ∈ NC ∧ y = (2c ↑c x))}) = {M})
35	2, 34	eqtrd 2385	1 ⊢ ((M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = {M})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ∅c0 3551  {csn 3738  0_cc0c 4375  ⟨cop 4562  {copab 4623   class class class wbr 4640   “ cima 4723   ‘cfv 4782  (class class class)co 5526   Clos1 cclos1 5873   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

This proof depends on 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 proof 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-fix 5741  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-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is used by:  nchoicelem9  6298  nchoicelem12  6301  nchoicelem15  6304  nchoicelem17  6306  nchoicelem19  6308