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Theorem qfto 6121

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 5708	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		brun 5193	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦))
3		df-br 5143	. . . . 5 ⊢ (𝑥(𝑅 ∪ ^◡𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 3473	. . . . . . 7 ⊢ 𝑥 ∈ V
5		vex 3473	. . . . . . 7 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5879	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
7	6	orbi2i 911	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥^◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
8	2, 3, 7	3bitr3i 301	. . . 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 1815	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
11		relxp 5690	. . 3 ⊢ Rel (𝐴 × 𝐵)
12		ssrel 5778	. . 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 3189	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
15	10, 13, 14	3bitr4i 303	1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∀wal 1532 ∈ wcel 2099 ∀wral 3056 ∪ cun 3942 ⊆ wss 3944 ⟨cop 4630 class class class wbr 5142 × cxp 5670 ^◡ccnv 5671 Rel wrel 5677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680

This theorem is referenced by: istsr2 18567 letsr 18576

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