New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  nnadjoinpw GIF version

Theorem nnadjoinpw 4521
 Description: Adjoining an element to a power class. Theorem X.1.40 of [Rosser] p. 530. (Contributed by SF, 27-Jan-2015.)
Assertion
Ref Expression
nnadjoinpw (((M Nn N Nn ) (A M X A) A N) → (A ∪ {X}) (N +c N))

Proof of Theorem nnadjoinpw
Dummy variables a b t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwadjoin 4119 . 2 (A ∪ {X}) = (A ∪ {a b Aa = (b ∪ {X})})
2 simp3 957 . . 3 (((M Nn N Nn ) (A M X A) A N) → A N)
3 simp1r 980 . . . 4 (((M Nn N Nn ) (A M X A) A N) → N Nn )
4 simp2r 982 . . . . 5 (((M Nn N Nn ) (A M X A) A N) → X A)
5 unipw 4117 . . . . . 6 A = A
65compleqi 3244 . . . . 5 A = ∼ A
74, 6syl6eleqr 2444 . . . 4 (((M Nn N Nn ) (A M X A) A N) → X A)
8 nnadjoin 4520 . . . 4 ((N Nn A N X A) → {a b Aa = (b ∪ {X})} N)
93, 2, 7, 8syl3anc 1182 . . 3 (((M Nn N Nn ) (A M X A) A N) → {a b Aa = (b ∪ {X})} N)
10 elcomplg 3218 . . . . . . . . 9 (X A → (X A ↔ ¬ X A))
1110ibi 232 . . . . . . . 8 (X A → ¬ X A)
124, 11syl 15 . . . . . . 7 (((M Nn N Nn ) (A M X A) A N) → ¬ X A)
13 snssg 3844 . . . . . . . 8 (X A → (X A ↔ {X} A))
144, 13syl 15 . . . . . . 7 (((M Nn N Nn ) (A M X A) A N) → (X A ↔ {X} A))
1512, 14mtbid 291 . . . . . 6 (((M Nn N Nn ) (A M X A) A N) → ¬ {X} A)
1615intnand 882 . . . . 5 (((M Nn N Nn ) (A M X A) A N) → ¬ (b A {X} A))
1716ralrimivw 2698 . . . 4 (((M Nn N Nn ) (A M X A) A N) → b A ¬ (b A {X} A))
18 disjr 3592 . . . . 5 ((A ∩ {a b Aa = (b ∪ {X})}) = t {a b Aa = (b ∪ {X})} ¬ t A)
19 eqeq1 2359 . . . . . . 7 (a = t → (a = (b ∪ {X}) ↔ t = (b ∪ {X})))
2019rexbidv 2635 . . . . . 6 (a = t → (b Aa = (b ∪ {X}) ↔ b At = (b ∪ {X})))
2120ralab 2997 . . . . 5 (t {a b Aa = (b ∪ {X})} ¬ t At(b At = (b ∪ {X}) → ¬ t A))
22 ralcom4 2877 . . . . . 6 (b At(t = (b ∪ {X}) → ¬ t A) ↔ tb A(t = (b ∪ {X}) → ¬ t A))
23 vex 2862 . . . . . . . . . 10 b V
24 snex 4111 . . . . . . . . . 10 {X} V
2523, 24unex 4106 . . . . . . . . 9 (b ∪ {X}) V
26 eleq1 2413 . . . . . . . . . 10 (t = (b ∪ {X}) → (t A ↔ (b ∪ {X}) A))
2726notbid 285 . . . . . . . . 9 (t = (b ∪ {X}) → (¬ t A ↔ ¬ (b ∪ {X}) A))
2825, 27ceqsalv 2885 . . . . . . . 8 (t(t = (b ∪ {X}) → ¬ t A) ↔ ¬ (b ∪ {X}) A)
2925elpw 3728 . . . . . . . . 9 ((b ∪ {X}) A ↔ (b ∪ {X}) A)
30 unss 3437 . . . . . . . . 9 ((b A {X} A) ↔ (b ∪ {X}) A)
3129, 30bitr4i 243 . . . . . . . 8 ((b ∪ {X}) A ↔ (b A {X} A))
3228, 31xchbinx 301 . . . . . . 7 (t(t = (b ∪ {X}) → ¬ t A) ↔ ¬ (b A {X} A))
3332ralbii 2638 . . . . . 6 (b At(t = (b ∪ {X}) → ¬ t A) ↔ b A ¬ (b A {X} A))
34 r19.23v 2730 . . . . . . 7 (b A(t = (b ∪ {X}) → ¬ t A) ↔ (b At = (b ∪ {X}) → ¬ t A))
3534albii 1566 . . . . . 6 (tb A(t = (b ∪ {X}) → ¬ t A) ↔ t(b At = (b ∪ {X}) → ¬ t A))
3622, 33, 353bitr3ri 267 . . . . 5 (t(b At = (b ∪ {X}) → ¬ t A) ↔ b A ¬ (b A {X} A))
3718, 21, 363bitri 262 . . . 4 ((A ∩ {a b Aa = (b ∪ {X})}) = b A ¬ (b A {X} A))
3817, 37sylibr 203 . . 3 (((M Nn N Nn ) (A M X A) A N) → (A ∩ {a b Aa = (b ∪ {X})}) = )
39 eladdci 4399 . . 3 ((A N {a b Aa = (b ∪ {X})} N (A ∩ {a b Aa = (b ∪ {X})}) = ) → (A ∪ {a b Aa = (b ∪ {X})}) (N +c N))
402, 9, 38, 39syl3anc 1182 . 2 (((M Nn N Nn ) (A M X A) A N) → (A ∪ {a b Aa = (b ∪ {X})}) (N +c N))
411, 40syl5eqel 2437 1 (((M Nn N Nn ) (A M X A) A N) → (A ∪ {X}) (N +c N))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  {csn 3737  ∪cuni 3891   Nn cnnc 4373   +c cplc 4375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  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-0c 4377  df-addc 4378  df-nnc 4379 This theorem is referenced by:  nnpweq  4523  sfindbl  4530
 Copyright terms: Public domain W3C validator