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Theorem qfto 5864

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 5486	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brun 5019	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦))
3		df-br 4969	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 3443	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 3443	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5646	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
7	6	orbi2i 907	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	2, 3, 7	3bitr3i 302	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
9	1, 8	imbi12i 352	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
10	9	2albii 1806	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
11		relxp 5468	. . 3 ⊢ Rel (𝐴 × 𝐵)
12		ssrel 5550	. . 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 3170	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
15	10, 13, 14	3bitr4i 304	1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 ∀wal 1523 ∈ wcel 2083 ∀wral 3107 ∪ cun 3863 ⊆ wss 3865 ⟨cop 4484 class class class wbr 4968 × cxp 5448 ^◡ccnv 5449 Rel wrel 5455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458

This theorem is referenced by: istsr2 17661 letsr 17670

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