Theorem opth 4603

Description: The ordered pair theorem. Two ordered pairs are equal iff their components are equal. (Contributed by SF, 2-Jan-2015.)

Assertion

Ref	Expression
opth	⊢ (⟨A, B⟩ = ⟨C, D⟩ ↔ (A = C ∧ B = D))

Proof of Theorem opth

Step	Hyp	Ref	Expression
1		proj1eq 4590	. . . 4 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → Proj1 ⟨A, B⟩ = Proj1 ⟨C, D⟩)
2		proj1op 4601	. . . 4 ⊢ Proj1 ⟨A, B⟩ = A
3		proj1op 4601	. . . 4 ⊢ Proj1 ⟨C, D⟩ = C
4	1, 2, 3	3eqtr3g 2408	. . 3 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → A = C)
5		proj2eq 4591	. . . 4 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → Proj2 ⟨A, B⟩ = Proj2 ⟨C, D⟩)
6		proj2op 4602	. . . 4 ⊢ Proj2 ⟨A, B⟩ = B
7		proj2op 4602	. . . 4 ⊢ Proj2 ⟨C, D⟩ = D
8	5, 6, 7	3eqtr3g 2408	. . 3 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → B = D)
9	4, 8	jca 518	. 2 ⊢ (⟨A, B⟩ = ⟨C, D⟩ → (A = C ∧ B = D))
10		opeq12 4581	. 2 ⊢ ((A = C ∧ B = D) → ⟨A, B⟩ = ⟨C, D⟩)
11	9, 10	impbii 180	1 ⊢ (⟨A, B⟩ = ⟨C, D⟩ ↔ (A = C ∧ B = D))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565

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

This theorem is referenced by:  eqvinop  4607  copsexg  4608  copsex4g  4611  opeqexb  4621  br1stg  4731  opelxp  4812  ralxpf  4828  brswap2  4861  dmsnopg  5067  cnvsn  5074  rnsnop  5076  dfxp2  5114  funsn  5148  fnasrn  5418  fsn  5433  opbr1st  5502  opbr2nd  5503  1stfo  5506  2ndfo  5507  swapf1o  5512  oprabid  5551  eloprabga  5579  brtxp  5784  fntxp  5805  dmpprod  5841  fnpprod  5844  addccan2nclem1  6264  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320