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Theorem qfto 5966

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 5572	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brun 5090	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦))
3		df-br 5040	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 3402	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 3402	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5736	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
7	6	orbi2i 913	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	2, 3, 7	3bitr3i 304	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
9	1, 8	imbi12i 354	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
10	9	2albii 1828	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
11		relxp 5554	. . 3 ⊢ Rel (𝐴 × 𝐵)
12		ssrel 5639	. . 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 3112	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∀wal 1541 ∈ wcel 2112 ∀wral 3051 ∪ cun 3851 ⊆ wss 3853 ⟨cop 4533 class class class wbr 5039 × cxp 5534 ^◡ccnv 5535 Rel wrel 5541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544

This theorem is referenced by: istsr2 18044 letsr 18053

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