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Theorem qfto 6111

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 5705	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brun 5192	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦))
3		df-br 5142	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 3477	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 3477	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5874	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
7	6	orbi2i 911	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	2, 3, 7	3bitr3i 300	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
9	1, 8	imbi12i 350	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
10	9	2albii 1822	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
11		relxp 5687	. . 3 ⊢ Rel (𝐴 × 𝐵)
12		ssrel 5774	. . 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 3193	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
15	10, 13, 14	3bitr4i 302	1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∀wal 1539 ∈ wcel 2106 ∀wral 3060 ∪ cun 3942 ⊆ wss 3944 ⟨cop 4628 class class class wbr 5141 × cxp 5667 ^◡ccnv 5668 Rel wrel 5674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677

This theorem is referenced by: istsr2 18519 letsr 18528

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