Theorem dich0 15291

Description: Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)

Assertion

Ref	Expression
dich0	⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem dich0

Step	Hyp	Ref	Expression
1		breq1 4065	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 ≤ 0 ↔ (𝑥 − 𝑦) ≤ 0))
2		breq2 4066	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥 − 𝑦)))
3	1, 2	orbi12d 797	. . . . 5 ⊢ (𝑧 = (𝑥 − 𝑦) → ((𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ((𝑥 − 𝑦) ≤ 0 ∨ 0 ≤ (𝑥 − 𝑦))))
4		simpl 109	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
5		resubcl 8378	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ)
6	5	adantl 277	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ)
7	3, 4, 6	rspcdva 2892	. . . 4 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) ≤ 0 ∨ 0 ≤ (𝑥 − 𝑦)))
8		simprl 529	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
9		simprr 531	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
10	8, 9	suble0d 8651	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) ≤ 0 ↔ 𝑥 ≤ 𝑦))
11	8, 9	subge0d 8650	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (0 ≤ (𝑥 − 𝑦) ↔ 𝑦 ≤ 𝑥))
12	10, 11	orbi12d 797	. . . 4 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (((𝑥 − 𝑦) ≤ 0 ∨ 0 ≤ (𝑥 − 𝑦)) ↔ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
13	7, 12	mpbid 147	. . 3 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
14	13	ralrimivva 2592	. 2 ⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
15		breq2 4066	. . . . 5 ⊢ (𝑦 = 0 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 0))
16		breq1 4065	. . . . 5 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑧 ↔ 0 ≤ 𝑧))
17	15, 16	orbi12d 797	. . . 4 ⊢ (𝑦 = 0 → ((𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧) ↔ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧)))
18		breq1 4065	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
19		breq2 4066	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑧))
20	18, 19	orbi12d 797	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧)))
21	20	ralbidv 2510	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧)))
22	21	rspccva 2886	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧))
23		0red 8115	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
24	17, 22, 23	rspcdva 2892	. . 3 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
25	24	ralrimiva 2583	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) → ∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
26	14, 25	impbii 126	1 ⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∨ wo 712   = wceq 1375   ∈ wcel 2180  ∀wral 2488   class class class wbr 4062  (class class class)co 5974  ℝcr 7966  0cc0 7967   ≤ cle 8150   − cmin 8285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288

This theorem is referenced by:  ivthdich  15292