Theorem qfto 6133

Description: A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Assertion

Ref	Expression
qfto	⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem qfto

Step	Hyp	Ref	Expression
1		opelxp 5694	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brun 5162	. . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦))
3		df-br 5109	. . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅))
4		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5872	. . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
7	6	orbi2i 926	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	2, 3, 7	3bitr3i 304	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
9	1, 8	imbi12i 353	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
10	9	2albii 1853	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
11		relxp 5676	. . 3 ⊢ Rel (𝐴 × 𝐵)
12		ssrel 5767	. . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅))))
13	11, 12	ax-mp 5	. 2 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ◡𝑅)))
14		r2al 3202	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
15	10, 13, 14	3bitr4i 306	1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∪ cun 4152  ⟨cop 4840   class class class wbr 5108   × cxp 5656  ^◡ccnv 5657  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666

This theorem is used by:  istsr2  18808  letsr  18817