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Theorem ncspw1eu 6159
 Description: Given a cardinal, there is a unique cardinal that contains the unit power class of its members. (Contributed by SF, 2-Mar-2015.)
Assertion
Ref Expression
ncspw1eu (A NC∃!x NC y A x = Nc 1y)
Distinct variable group:   x,A,y

Proof of Theorem ncspw1eu
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nulnnc 6118 . . . . . . 7 ¬ NC
2 eleq1 2413 . . . . . . 7 (A = → (A NC NC ))
31, 2mtbiri 294 . . . . . 6 (A = → ¬ A NC )
43necon2ai 2561 . . . . 5 (A NCA)
5 n0 3559 . . . . 5 (Ay y A)
64, 5sylib 188 . . . 4 (A NCy y A)
7 vex 2862 . . . . . . . . . 10 y V
87pw1ex 4303 . . . . . . . . 9 1y V
98ncelncsi 6121 . . . . . . . 8 Nc 1y NC
10 eqid 2353 . . . . . . . 8 Nc 1y = Nc 1y
11 eqeq1 2359 . . . . . . . . 9 (x = Nc 1y → (x = Nc 1yNc 1y = Nc 1y))
1211rspcev 2955 . . . . . . . 8 (( Nc 1y NC Nc 1y = Nc 1y) → x NC x = Nc 1y)
139, 10, 12mp2an 653 . . . . . . 7 x NC x = Nc 1y
1413jctr 526 . . . . . 6 (y A → (y A x NC x = Nc 1y))
1514a1i 10 . . . . 5 (A NC → (y A → (y A x NC x = Nc 1y)))
1615eximdv 1622 . . . 4 (A NC → (y y Ay(y A x NC x = Nc 1y)))
176, 16mpd 14 . . 3 (A NCy(y A x NC x = Nc 1y))
18 rexcom 2772 . . . 4 (x NC y A x = Nc 1yy A x NC x = Nc 1y)
19 df-rex 2620 . . . 4 (y A x NC x = Nc 1yy(y A x NC x = Nc 1y))
2018, 19bitri 240 . . 3 (x NC y A x = Nc 1yy(y A x NC x = Nc 1y))
2117, 20sylibr 203 . 2 (A NCx NC y A x = Nc 1y)
22 reeanv 2778 . . . 4 (y A w A (x = Nc 1y z = Nc 1w) ↔ (y A x = Nc 1y w A z = Nc 1w))
23 ncseqnc 6128 . . . . . . . . . . . 12 (A NC → (A = Nc yy A))
2423biimpar 471 . . . . . . . . . . 11 ((A NC y A) → A = Nc y)
2524adantrr 697 . . . . . . . . . 10 ((A NC (y A w A)) → A = Nc y)
26 ncseqnc 6128 . . . . . . . . . . . 12 (A NC → (A = Nc ww A))
2726biimpar 471 . . . . . . . . . . 11 ((A NC w A) → A = Nc w)
2827adantrl 696 . . . . . . . . . 10 ((A NC (y A w A)) → A = Nc w)
2925, 28eqtr3d 2387 . . . . . . . . 9 ((A NC (y A w A)) → Nc y = Nc w)
307ncpw1 6152 . . . . . . . . 9 ( Nc y = Nc wNc 1y = Nc 1w)
3129, 30sylib 188 . . . . . . . 8 ((A NC (y A w A)) → Nc 1y = Nc 1w)
32313adant2 974 . . . . . . 7 ((A NC (x NC z NC ) (y A w A)) → Nc 1y = Nc 1w)
33 eqeq2 2362 . . . . . . . . 9 ( Nc 1y = Nc 1w → (x = Nc 1yx = Nc 1w))
3433anbi1d 685 . . . . . . . 8 ( Nc 1y = Nc 1w → ((x = Nc 1y z = Nc 1w) ↔ (x = Nc 1w z = Nc 1w)))
35 eqtr3 2372 . . . . . . . 8 ((x = Nc 1w z = Nc 1w) → x = z)
3634, 35syl6bi 219 . . . . . . 7 ( Nc 1y = Nc 1w → ((x = Nc 1y z = Nc 1w) → x = z))
3732, 36syl 15 . . . . . 6 ((A NC (x NC z NC ) (y A w A)) → ((x = Nc 1y z = Nc 1w) → x = z))
38373expa 1151 . . . . 5 (((A NC (x NC z NC )) (y A w A)) → ((x = Nc 1y z = Nc 1w) → x = z))
3938rexlimdvva 2745 . . . 4 ((A NC (x NC z NC )) → (y A w A (x = Nc 1y z = Nc 1w) → x = z))
4022, 39syl5bir 209 . . 3 ((A NC (x NC z NC )) → ((y A x = Nc 1y w A z = Nc 1w) → x = z))
4140ralrimivva 2706 . 2 (A NCx NC z NC ((y A x = Nc 1y w A z = Nc 1w) → x = z))
42 eqeq1 2359 . . . . 5 (x = z → (x = Nc 1yz = Nc 1y))
4342rexbidv 2635 . . . 4 (x = z → (y A x = Nc 1yy A z = Nc 1y))
44 pw1eq 4143 . . . . . . 7 (y = w1y = 1w)
4544nceqd 6110 . . . . . 6 (y = wNc 1y = Nc 1w)
4645eqeq2d 2364 . . . . 5 (y = w → (z = Nc 1yz = Nc 1w))
4746cbvrexv 2836 . . . 4 (y A z = Nc 1yw A z = Nc 1w)
4843, 47syl6bb 252 . . 3 (x = z → (y A x = Nc 1yw A z = Nc 1w))
4948reu4 3030 . 2 (∃!x NC y A x = Nc 1y ↔ (x NC y A x = Nc 1y x NC z NC ((y A x = Nc 1y w A z = Nc 1w) → x = z)))
5021, 41, 49sylanbrc 645 1 (A NC∃!x NC y A x = Nc 1y)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616  ∅c0 3550  ℘1cpw1 4135   NC cncs 6088   Nc cnc 6091 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 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-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-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101 This theorem is referenced by:  tccl  6160  eqtc  6161
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