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Theorem qfto 4998
Description: A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
qfto ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑦𝑅𝑥))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem qfto
StepHypRef Expression
1 opelxp 4639 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2 brun 4038 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ (𝑥𝑅𝑦𝑥𝑅𝑦))
3 df-br 3988 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))
4 vex 2733 . . . . . . 7 𝑥 ∈ V
5 vex 2733 . . . . . . 7 𝑦 ∈ V
64, 5brcnv 4792 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
76orbi2i 757 . . . . 5 ((𝑥𝑅𝑦𝑥𝑅𝑦) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
82, 3, 73bitr3i 209 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
91, 8imbi12i 238 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
1092albii 1464 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
11 relxp 4718 . . 3 Rel (𝐴 × 𝐵)
12 ssrel 4697 . . 3 (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))))
1311, 12ax-mp 5 . 2 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)))
14 r2al 2489 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑦𝑅𝑥)))
1510, 13, 143bitr4i 211 1 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑦𝑅𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703  wal 1346  wcel 2141  wral 2448  cun 3119  wss 3121  cop 3584   class class class wbr 3987   × cxp 4607  ccnv 4608  Rel wrel 4614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617
This theorem is referenced by: (None)
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