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Theorem xpdom1 6309
 Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
Hypothesis
Ref Expression
xpdom1.2 𝐶 ∈ V
Assertion
Ref Expression
xpdom1 (𝐴𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))

Proof of Theorem xpdom1
StepHypRef Expression
1 xpdom1.2 . 2 𝐶 ∈ V
2 xpdom1g 6307 . 2 ((𝐶 ∈ V ∧ 𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
31, 2mpan 400 1 (𝐴𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764   × cxp 4343   ≼ cdom 6220 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1st 5767  df-2nd 5768  df-en 6222  df-dom 6223 This theorem is referenced by: (None)
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