Theorem dich0 15504

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 4111	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 ≤ 0 ↔ (𝑥 − 𝑦) ≤ 0))
2		breq2 4112	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥 − 𝑦)))
3	1, 2	orbi12d 801	. . . . 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 8533	. . . . . 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 2925	. . . 4 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) ≤ 0 ∨ 0 ≤ (𝑥 − 𝑦)))
8		simprl 531	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
9		simprr 533	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
10	8, 9	suble0d 8806	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) ≤ 0 ↔ 𝑥 ≤ 𝑦))
11	8, 9	subge0d 8805	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (0 ≤ (𝑥 − 𝑦) ↔ 𝑦 ≤ 𝑥))
12	10, 11	orbi12d 801	. . . 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 2624	. 2 ⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
15		breq2 4112	. . . . 5 ⊢ (𝑦 = 0 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 0))
16		breq1 4111	. . . . 5 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑧 ↔ 0 ≤ 𝑧))
17	15, 16	orbi12d 801	. . . 4 ⊢ (𝑦 = 0 → ((𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧) ↔ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧)))
18		breq1 4111	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
19		breq2 4112	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑧))
20	18, 19	orbi12d 801	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧)))
21	20	ralbidv 2542	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧)))
22	21	rspccva 2919	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧))
23		0red 8271	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
24	17, 22, 23	rspcdva 2925	. . 3 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
25	24	ralrimiva 2615	. 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 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520   class class class wbr 4108  (class class class)co 6049  ℝcr 8122  0cc0 8123   ≤ cle 8305   − cmin 8440

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443

This theorem is referenced by:  ivthdich  15505