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Theorem nenpw1pwlem2 6085
 Description: Lemma for nenpw1pw 6086. Establish the main theorem with an extra hypothesis. (Contributed by SF, 10-Mar-2015.)
Hypothesis
Ref Expression
nenpw1pwlem2.1 S = {x A ¬ x (r ‘{x})}
Assertion
Ref Expression
nenpw1pwlem2 ¬ 1AA
Distinct variable group:   A,r,x
Allowed substitution hints:   S(x,r)

Proof of Theorem nenpw1pwlem2
Dummy variables u y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm5.19 349 . . . . 5 ¬ (y S ↔ ¬ y S)
21a1i 10 . . . 4 (y A → ¬ (y S ↔ ¬ y S))
32nrex 2716 . . 3 ¬ y A (y S ↔ ¬ y S)
43nex 1555 . 2 ¬ ry A (y S ↔ ¬ y S)
5 bren 6030 . . 3 (1AAr r:1A1-1-ontoA)
6 f1odm 5290 . . . . . . . . 9 (r:1A1-1-ontoA → dom r = 1A)
7 vex 2862 . . . . . . . . . 10 r V
87dmex 5106 . . . . . . . . 9 dom r V
96, 8syl6eqelr 2442 . . . . . . . 8 (r:1A1-1-ontoA1A V)
10 pw1exb 4326 . . . . . . . 8 (1A V ↔ A V)
119, 10sylib 188 . . . . . . 7 (r:1A1-1-ontoAA V)
12 nenpw1pwlem2.1 . . . . . . . 8 S = {x A ¬ x (r ‘{x})}
1312nenpw1pwlem1 6084 . . . . . . 7 (A V → S V)
1411, 13syl 15 . . . . . 6 (r:1A1-1-ontoAS V)
15 ssrab2 3351 . . . . . . . 8 {x A ¬ x (r ‘{x})} A
1612, 15eqsstri 3301 . . . . . . 7 S A
17 elpwg 3729 . . . . . . 7 (S V → (S AS A))
1816, 17mpbiri 224 . . . . . 6 (S V → S A)
1914, 18syl 15 . . . . 5 (r:1A1-1-ontoAS A)
20 f1ofo 5293 . . . . . . . 8 (r:1A1-1-ontoAr:1AontoA)
21 forn 5272 . . . . . . . 8 (r:1AontoA → ran r = A)
2220, 21syl 15 . . . . . . 7 (r:1A1-1-ontoA → ran r = A)
2322eleq2d 2420 . . . . . 6 (r:1A1-1-ontoA → (S ran rS A))
24 elrn 4896 . . . . . . 7 (S ran ru urS)
25 breldm 4911 . . . . . . . . . . . 12 (urSu dom r)
2625adantl 452 . . . . . . . . . . 11 ((r:1A1-1-ontoA urS) → u dom r)
276adantr 451 . . . . . . . . . . 11 ((r:1A1-1-ontoA urS) → dom r = 1A)
2826, 27eleqtrd 2429 . . . . . . . . . 10 ((r:1A1-1-ontoA urS) → u 1A)
29 elpw1 4144 . . . . . . . . . . 11 (u 1Ay A u = {y})
30 breq1 4642 . . . . . . . . . . . . . . . 16 (u = {y} → (urS ↔ {y}rS))
31303anbi2d 1257 . . . . . . . . . . . . . . 15 (u = {y} → ((r:1A1-1-ontoA urS y A) ↔ (r:1A1-1-ontoA {y}rS y A)))
32 id 19 . . . . . . . . . . . . . . . . . . 19 (x = yx = y)
33 sneq 3744 . . . . . . . . . . . . . . . . . . . 20 (x = y → {x} = {y})
3433fveq2d 5332 . . . . . . . . . . . . . . . . . . 19 (x = y → (r ‘{x}) = (r ‘{y}))
3532, 34eleq12d 2421 . . . . . . . . . . . . . . . . . 18 (x = y → (x (r ‘{x}) ↔ y (r ‘{y})))
3635notbid 285 . . . . . . . . . . . . . . . . 17 (x = y → (¬ x (r ‘{x}) ↔ ¬ y (r ‘{y})))
3736, 12elrab2 2996 . . . . . . . . . . . . . . . 16 (y S ↔ (y A ¬ y (r ‘{y})))
38 simp3 957 . . . . . . . . . . . . . . . . . 18 ((r:1A1-1-ontoA {y}rS y A) → y A)
3938biantrurd 494 . . . . . . . . . . . . . . . . 17 ((r:1A1-1-ontoA {y}rS y A) → (¬ y (r ‘{y}) ↔ (y A ¬ y (r ‘{y}))))
40 simp2 956 . . . . . . . . . . . . . . . . . . . 20 ((r:1A1-1-ontoA {y}rS y A) → {y}rS)
41 f1ofn 5288 . . . . . . . . . . . . . . . . . . . . . 22 (r:1A1-1-ontoAr Fn 1A)
42413ad2ant1 976 . . . . . . . . . . . . . . . . . . . . 21 ((r:1A1-1-ontoA {y}rS y A) → r Fn 1A)
43 snelpw1 4146 . . . . . . . . . . . . . . . . . . . . . . 23 ({y} 1Ay A)
4443biimpri 197 . . . . . . . . . . . . . . . . . . . . . 22 (y A → {y} 1A)
45443ad2ant3 978 . . . . . . . . . . . . . . . . . . . . 21 ((r:1A1-1-ontoA {y}rS y A) → {y} 1A)
46 fnbrfvb 5358 . . . . . . . . . . . . . . . . . . . . 21 ((r Fn 1A {y} 1A) → ((r ‘{y}) = S ↔ {y}rS))
4742, 45, 46syl2anc 642 . . . . . . . . . . . . . . . . . . . 20 ((r:1A1-1-ontoA {y}rS y A) → ((r ‘{y}) = S ↔ {y}rS))
4840, 47mpbird 223 . . . . . . . . . . . . . . . . . . 19 ((r:1A1-1-ontoA {y}rS y A) → (r ‘{y}) = S)
4948eleq2d 2420 . . . . . . . . . . . . . . . . . 18 ((r:1A1-1-ontoA {y}rS y A) → (y (r ‘{y}) ↔ y S))
5049notbid 285 . . . . . . . . . . . . . . . . 17 ((r:1A1-1-ontoA {y}rS y A) → (¬ y (r ‘{y}) ↔ ¬ y S))
5139, 50bitr3d 246 . . . . . . . . . . . . . . . 16 ((r:1A1-1-ontoA {y}rS y A) → ((y A ¬ y (r ‘{y})) ↔ ¬ y S))
5237, 51syl5bb 248 . . . . . . . . . . . . . . 15 ((r:1A1-1-ontoA {y}rS y A) → (y S ↔ ¬ y S))
5331, 52syl6bi 219 . . . . . . . . . . . . . 14 (u = {y} → ((r:1A1-1-ontoA urS y A) → (y S ↔ ¬ y S)))
5453com12 27 . . . . . . . . . . . . 13 ((r:1A1-1-ontoA urS y A) → (u = {y} → (y S ↔ ¬ y S)))
55543expa 1151 . . . . . . . . . . . 12 (((r:1A1-1-ontoA urS) y A) → (u = {y} → (y S ↔ ¬ y S)))
5655reximdva 2726 . . . . . . . . . . 11 ((r:1A1-1-ontoA urS) → (y A u = {y} → y A (y S ↔ ¬ y S)))
5729, 56syl5bi 208 . . . . . . . . . 10 ((r:1A1-1-ontoA urS) → (u 1Ay A (y S ↔ ¬ y S)))
5828, 57mpd 14 . . . . . . . . 9 ((r:1A1-1-ontoA urS) → y A (y S ↔ ¬ y S))
5958ex 423 . . . . . . . 8 (r:1A1-1-ontoA → (urSy A (y S ↔ ¬ y S)))
6059exlimdv 1636 . . . . . . 7 (r:1A1-1-ontoA → (u urSy A (y S ↔ ¬ y S)))
6124, 60syl5bi 208 . . . . . 6 (r:1A1-1-ontoA → (S ran ry A (y S ↔ ¬ y S)))
6223, 61sylbird 226 . . . . 5 (r:1A1-1-ontoA → (S Ay A (y S ↔ ¬ y S)))
6319, 62mpd 14 . . . 4 (r:1A1-1-ontoAy A (y S ↔ ¬ y S))
6463eximi 1576 . . 3 (r r:1A1-1-ontoAry A (y S ↔ ¬ y S))
655, 64sylbi 187 . 2 (1AAry A (y S ↔ ¬ y S))
664, 65mto 167 1 ¬ 1AA
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {crab 2618  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ℘1cpw1 4135   class class class wbr 4639  dom cdm 4772  ran crn 4773   Fn wfn 4776  –onto→wfo 4779  –1-1-onto→wf1o 4780   ‘cfv 4781   ≈ cen 6028 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-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-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-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 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-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-fullfun 5768  df-en 6029 This theorem is referenced by:  nenpw1pw  6086
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