qfto - Intuitionistic Logic Explorer

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Theorem qfto 4637

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 4297	. . . 4 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
2		brun 3780	. . . . 5 ⊢ (x(𝑅 ∪ ^◡𝑅)y ↔ (x𝑅y ∨ x^◡𝑅y))
3		df-br 3735	. . . . 5 ⊢ (x(𝑅 ∪ ^◡𝑅)y ↔ ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅))
4		vex 2534	. . . . . . 7 ⊢ x ∈ V
5		vex 2534	. . . . . . 7 ⊢ y ∈ V
6	4, 5	brcnv 4441	. . . . . 6 ⊢ (x^◡𝑅y ↔ y𝑅x)
7	6	orbi2i 666	. . . . 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 1336	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (A × B) → ⟨x, y⟩ ∈ (𝑅 ∪ ^◡𝑅)) ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → (x𝑅y ∨ y𝑅x)))
11		relxp 4370	. . 3 ⊢ Rel (A × B)
12		ssrel 4351	. . 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 2317	. 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 616 ∀wal 1224 ∈ wcel 1370 ∀wral 2280 ∪ cun 2888 ⊆ wss 2890 ⟨cop 3349 class class class wbr 3734 × cxp 4266 ^◡ccnv 4267 Rel wrel 4273

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 617 ax-5 1312 ax-7 1313 ax-gen 1314 ax-ie1 1359 ax-ie2 1360 ax-8 1372 ax-10 1373 ax-11 1374 ax-i12 1375 ax-bnd 1376 ax-4 1377 ax-14 1382 ax-17 1396 ax-i9 1400 ax-ial 1405 ax-i5r 1406 ax-ext 2000 ax-sep 3845 ax-pow 3897 ax-pr 3914

This theorem depends on definitions: df-bi 110 df-3an 873 df-tru 1229 df-nf 1326 df-sb 1624 df-eu 1881 df-mo 1882 df-clab 2005 df-cleq 2011 df-clel 2014 df-nfc 2145 df-ral 2285 df-rex 2286 df-v 2533 df-un 2895 df-in 2897 df-ss 2904 df-pw 3332 df-sn 3352 df-pr 3353 df-op 3355 df-br 3735 df-opab 3789 df-xp 4274 df-rel 4275 df-cnv 4276

This theorem is referenced by: (None)

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