Theorem tc11 6229

Description: Cardinal T is one-to-one. Based on theorem 2.4 of [Specker] p. 972. (Contributed by SF, 10-Mar-2015.)

Assertion

Ref	Expression
tc11	⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M = Tc N ↔ M = N))

Proof of Theorem tc11

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6120	. . . 4 ⊢ (M ∈ NC ↔ ∃x M = Nc x)
2		elncs 6120	. . . 4 ⊢ (N ∈ NC ↔ ∃y N = Nc y)
3	1, 2	anbi12i 678	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) ↔ (∃x M = Nc x ∧ ∃y N = Nc y))
4		eeanv 1913	. . 3 ⊢ (∃x∃y(M = Nc x ∧ N = Nc y) ↔ (∃x M = Nc x ∧ ∃y N = Nc y))
5	3, 4	bitr4i 243	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) ↔ ∃x∃y(M = Nc x ∧ N = Nc y))
6		vex 2863	. . . . . . 7 ⊢ x ∈ V
7	6	tcnc 6226	. . . . . 6 ⊢ Tc Nc x = Nc ℘1x
8		vex 2863	. . . . . . 7 ⊢ y ∈ V
9	8	tcnc 6226	. . . . . 6 ⊢ Tc Nc y = Nc ℘1y
10	7, 9	eqeq12i 2366	. . . . 5 ⊢ ( Tc Nc x = Tc Nc y ↔ Nc ℘1x = Nc ℘1y)
11		enpw1 6063	. . . . . 6 ⊢ (x ≈ y ↔ ℘1x ≈ ℘1y)
12	6	eqnc 6128	. . . . . 6 ⊢ ( Nc x = Nc y ↔ x ≈ y)
13	6	pw1ex 4304	. . . . . . 7 ⊢ ℘1x ∈ V
14	13	eqnc 6128	. . . . . 6 ⊢ ( Nc ℘1x = Nc ℘1y ↔ ℘1x ≈ ℘1y)
15	11, 12, 14	3bitr4ri 269	. . . . 5 ⊢ ( Nc ℘1x = Nc ℘1y ↔ Nc x = Nc y)
16	10, 15	bitri 240	. . . 4 ⊢ ( Tc Nc x = Tc Nc y ↔ Nc x = Nc y)
17		tceq 6159	. . . . . 6 ⊢ (M = Nc x → Tc M = Tc Nc x)
18		tceq 6159	. . . . . 6 ⊢ (N = Nc y → Tc N = Tc Nc y)
19	17, 18	eqeqan12d 2368	. . . . 5 ⊢ ((M = Nc x ∧ N = Nc y) → ( Tc M = Tc N ↔ Tc Nc x = Tc Nc y))
20		eqeq12 2365	. . . . 5 ⊢ ((M = Nc x ∧ N = Nc y) → (M = N ↔ Nc x = Nc y))
21	19, 20	bibi12d 312	. . . 4 ⊢ ((M = Nc x ∧ N = Nc y) → (( Tc M = Tc N ↔ M = N) ↔ ( Tc Nc x = Tc Nc y ↔ Nc x = Nc y)))
22	16, 21	mpbiri 224	. . 3 ⊢ ((M = Nc x ∧ N = Nc y) → ( Tc M = Tc N ↔ M = N))
23	22	exlimivv 1635	. 2 ⊢ (∃x∃y(M = Nc x ∧ N = Nc y) → ( Tc M = Tc N ↔ M = N))
24	5, 23	sylbi 187	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M = Tc N ↔ M = N))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ℘₁cpw1 4136   class class class wbr 4640   ≈ cen 6029   NC cncs 6089   Nc cnc 6092   T_c ctc 6094

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104

This theorem is referenced by:  tlecg  6231