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Theorem rngoideu 25217
 Description: The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1
ringi.2
ringi.3
Assertion
Ref Expression
rngoideu
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem rngoideu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5
2 ringi.2 . . . . 5
3 ringi.3 . . . . 5
41, 2, 3rngoi 25213 . . . 4
5 simprr 756 . . . 4
64, 5syl 16 . . 3
7 simpl 457 . . . . . . . 8
87ralimi 2860 . . . . . . 7
9 oveq2 6303 . . . . . . . . 9
10 id 22 . . . . . . . . 9
119, 10eqeq12d 2489 . . . . . . . 8
1211rspcv 3215 . . . . . . 7
138, 12syl5 32 . . . . . 6
14 simpr 461 . . . . . . . 8
1514ralimi 2860 . . . . . . 7
16 oveq1 6302 . . . . . . . . 9
17 id 22 . . . . . . . . 9
1816, 17eqeq12d 2489 . . . . . . . 8
1918rspcv 3215 . . . . . . 7
2015, 19syl5 32 . . . . . 6
2113, 20im2anan9r 834 . . . . 5
22 eqtr2 2494 . . . . . 6
2322eqcomd 2475 . . . . 5
2421, 23syl6 33 . . . 4
2524rgen2a 2894 . . 3
266, 25jctir 538 . 2
27 oveq1 6302 . . . . . 6
2827eqeq1d 2469 . . . . 5
29 oveq2 6303 . . . . . 6
3029eqeq1d 2469 . . . . 5
3128, 30anbi12d 710 . . . 4
3231ralbidv 2906 . . 3
3332reu4 3302 . 2
3426, 33sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  wrex 2818  wreu 2819   cxp 5003   crn 5006  wf 5590  cfv 5594  (class class class)co 6295  c1st 6793  c2nd 6794  cablo 25114  crngo 25208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-rngo 25209 This theorem is referenced by: (None)
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