qfto - Intuitionistic Logic Explorer

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Theorem qfto 4657

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 4317	. . . 4 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
2		brun 3801	. . . . 5 ⊢ (x(𝑅 ∪ ^◡𝑅)y ↔ (x𝑅y ∨ x^◡𝑅y))
3		df-br 3756	. . . . 5 ⊢ (x(𝑅 ∪ ^◡𝑅)y ↔ ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 2554	. . . . . . 7 ⊢ x ∈ V
5		vex 2554	. . . . . . 7 ⊢ y ∈ V
6	4, 5	brcnv 4461	. . . . . 6 ⊢ (x^◡𝑅y ↔ y𝑅x)
7	6	orbi2i 678	. . . . 5 ⊢ ((x𝑅y ∨ x^◡𝑅y) ↔ (x𝑅y ∨ y𝑅x))
8	2, 3, 7	3bitr3i 199	. . . 4 ⊢ (⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅) ↔ (x𝑅y ∨ y𝑅x))
9	1, 8	imbi12i 228	. . 3 ⊢ ((⟨x, y⟩ ∈ (A × B) → ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x)))
10	9	2albii 1357	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (A × B) → ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x)))
11		relxp 4390	. . 3 ⊢ Rel (A × B)
12		ssrel 4371	. . 3 ⊢ (Rel (A × B) → ((A × B) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀x∀y(⟨x, y⟩ ∈ (A × B) → ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅))))
13	11, 12	ax-mp 7	. 2 ⊢ ((A × B) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀x∀y(⟨x, y⟩ ∈ (A × B) → ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅)))
14		r2al 2337	. 2 ⊢ (∀x ∈ A ∀y ∈ B (x𝑅y ∨ y𝑅x) ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x)))
15	10, 13, 14	3bitr4i 201	1 ⊢ ((A × B) ⊆ (𝑅 ∪ ^◡𝑅) ↔ ∀x ∈ A ∀y ∈ B (x𝑅y ∨ y𝑅x))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 ∀wal 1240 ∈ wcel 1390 ∀wral 2300 ∪ cun 2909 ⊆ wss 2911 ⟨cop 3370 class class class wbr 3755 × cxp 4286 ^◡ccnv 4287 Rel wrel 4293

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296

This theorem is referenced by: (None)

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