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 = B ↔ ∃x ∈ A B = Nc ℘1x))

Distinct variable groups:   x,A   x,B

Proof of Theorem eqtc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 447	. . . 4 ⊢ ((A ∈ NC ∧ Tc A = B) → Tc A = B)
2		tccl 6160	. . . . 5 ⊢ (A ∈ NC → Tc A ∈ NC )
3	2	adantr 451	. . . 4 ⊢ ((A ∈ NC ∧ Tc A = B) → Tc A ∈ NC )
4	1, 3	eqeltrrd 2428	. . 3 ⊢ ((A ∈ NC ∧ Tc A = B) → B ∈ NC )
5	4	ex 423	. 2 ⊢ (A ∈ NC → ( Tc A = B → B ∈ NC ))
6		vex 2862	. . . . . . 7 ⊢ x ∈ V
7	6	pw1ex 4303	. . . . . 6 ⊢ ℘1x ∈ V
8	7	ncelncsi 6121	. . . . 5 ⊢ Nc ℘1x ∈ NC
9		eleq1 2413	. . . . 5 ⊢ (B = Nc ℘1x → (B ∈ NC ↔ Nc ℘1x ∈ NC ))
10	8, 9	mpbiri 224	. . . 4 ⊢ (B = Nc ℘1x → B ∈ NC )
11	10	rexlimivw 2734	. . 3 ⊢ (∃x ∈ A B = Nc ℘1x → B ∈ NC )
12	11	a1i 10	. 2 ⊢ (A ∈ NC → (∃x ∈ A B = Nc ℘1x → B ∈ NC ))
13		ncspw1eu 6159	. . . . . 6 ⊢ (A ∈ NC → ∃!y ∈ NC ∃x ∈ A y = Nc ℘1x)
14		eqeq1 2359	. . . . . . . 8 ⊢ (y = B → (y = Nc ℘1x ↔ B = Nc ℘1x))
15	14	rexbidv 2635	. . . . . . 7 ⊢ (y = B → (∃x ∈ A y = Nc ℘1x ↔ ∃x ∈ A B = Nc ℘1x))
16	15	reiota2 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))
17	13, 16	sylan2 460	. . . . 5 ⊢ ((B ∈ NC ∧ A ∈ NC ) → (∃x ∈ A B = Nc ℘1x ↔ (℩y(y ∈ NC ∧ ∃x ∈ A y = Nc ℘1x)) = B))
18	17	ancoms 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))
20	19	eqeq1i 2360	. . . 4 ⊢ ( Tc A = B ↔ (℩y(y ∈ NC ∧ ∃x ∈ A y = Nc ℘1x)) = B)
21	18, 20	syl6rbbr 255	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ( Tc A = B ↔ ∃x ∈ A B = Nc ℘1x))
22	21	ex 423	. 2 ⊢ (A ∈ NC → (B ∈ NC → ( Tc A = B ↔ ∃x ∈ A B = Nc ℘1x)))
23	5, 12, 22	pm5.21ndd 343	1 ⊢ (A ∈ NC → ( Tc A = B ↔ ∃x ∈ A B = Nc ℘1x))

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   T_c ctc 6093

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  df-tc 6103

This theorem is referenced by:  pw1eltc  6162