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Theorem opelxp1 4320
 Description: The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp1 (⟨A, B (𝐶 × 𝐷) → A 𝐶)

Proof of Theorem opelxp1
StepHypRef Expression
1 opelxp 4317 . 2 (⟨A, B (𝐶 × 𝐷) ↔ (A 𝐶 B 𝐷))
21simplbi 259 1 (⟨A, B (𝐶 × 𝐷) → A 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  ⟨cop 3370   × cxp 4286 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294 This theorem is referenced by:  otelxp1  4322  dmxpss  4696  nfvres  5149  ressnop0  5287  swoord1  6071  swoord2  6072
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