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Theorem ideq2 34421
Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.)
Assertion
Ref Expression
ideq2 (𝐴𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem ideq2
StepHypRef Expression
1 brid 34420 . 2 (𝐴 I 𝐵𝐵 I 𝐴)
2 ideqg 5430 . . 3 (𝐴𝑉 → (𝐵 I 𝐴𝐵 = 𝐴))
3 eqcom 2768 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
42, 3syl6bb 276 . 2 (𝐴𝑉 → (𝐵 I 𝐴𝐴 = 𝐵))
51, 4syl5bb 272 1 (𝐴𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2140   class class class wbr 4805   I cid 5174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275
This theorem is referenced by:  br1cossinidres  34541  br1cossxrnidres  34543
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