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Theorem eqtc 6161
 Description: The defining property of the cardinal T operation. (Contributed by SF, 2-Mar-2015.)
Assertion
Ref Expression
eqtc (A NC → ( Tc A = Bx A B = Nc 1x))
Distinct variable groups:   x,A   x,B

Proof of Theorem eqtc
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4 ((A NC Tc A = B) → Tc A = B)
2 tccl 6160 . . . . 5 (A NCTc A NC )
32adantr 451 . . . 4 ((A NC Tc A = B) → Tc A NC )
41, 3eqeltrrd 2428 . . 3 ((A NC Tc A = B) → B NC )
54ex 423 . 2 (A NC → ( Tc A = BB NC ))
6 vex 2862 . . . . . . 7 x V
76pw1ex 4303 . . . . . 6 1x V
87ncelncsi 6121 . . . . 5 Nc 1x NC
9 eleq1 2413 . . . . 5 (B = Nc 1x → (B NCNc 1x NC ))
108, 9mpbiri 224 . . . 4 (B = Nc 1xB NC )
1110rexlimivw 2734 . . 3 (x A B = Nc 1xB NC )
1211a1i 10 . 2 (A NC → (x A B = Nc 1xB NC ))
13 ncspw1eu 6159 . . . . . 6 (A NC∃!y NC x A y = Nc 1x)
14 eqeq1 2359 . . . . . . . 8 (y = B → (y = Nc 1xB = Nc 1x))
1514rexbidv 2635 . . . . . . 7 (y = B → (x A y = Nc 1xx A B = Nc 1x))
1615reiota2 4368 . . . . . 6 ((B NC ∃!y NC x A y = Nc 1x) → (x A B = Nc 1x ↔ (℩y(y NC x A y = Nc 1x)) = B))
1713, 16sylan2 460 . . . . 5 ((B NC A NC ) → (x A B = Nc 1x ↔ (℩y(y NC x A y = Nc 1x)) = B))
1817ancoms 439 . . . 4 ((A NC B NC ) → (x A B = Nc 1x ↔ (℩y(y NC x A y = Nc 1x)) = B))
19 df-tc 6103 . . . . 5 Tc A = (℩y(y NC x A y = Nc 1x))
2019eqeq1i 2360 . . . 4 ( Tc A = B ↔ (℩y(y NC x A y = Nc 1x)) = B)
2118, 20syl6rbbr 255 . . 3 ((A NC B NC ) → ( Tc A = Bx A B = Nc 1x))
2221ex 423 . 2 (A NC → (B NC → ( Tc A = Bx A B = Nc 1x)))
235, 12, 22pm5.21ndd 343 1 (A NC → ( Tc A = Bx A B = Nc 1x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∃!wreu 2616  ℘1cpw1 4135  ℩cio 4337   NC cncs 6088   Nc cnc 6091   Tc ctc 6093 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-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  df-tc 6103 This theorem is referenced by:  pw1eltc  6162
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