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Theorem elxp2 4363
 Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
Assertion
Ref Expression
elxp2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp2
StepHypRef Expression
1 df-rex 2312 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)))
2 r19.42v 2467 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3 an13 497 . . . . 5 ((𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
43exbii 1496 . . . 4 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
51, 2, 43bitr3i 199 . . 3 ((𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
65exbii 1496 . 2 (∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
7 df-rex 2312 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8 elxp 4362 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
96, 7, 83bitr4ri 202 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   × cxp 4343 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-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 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351 This theorem is referenced by:  opelxp  4374  xpiundi  4398  xpiundir  4399  ssrel2  4430  f1o2ndf1  5849  xpdom2  6305  elreal  6905
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