Theorem dich0 15566

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 4114	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 ≤ 0 ↔ (𝑥 − 𝑦) ≤ 0))
2		breq2 4115	. . . . . 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 8542	. . . . . 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 2928	. . . 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 8815	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) ≤ 0 ↔ 𝑥 ≤ 𝑦))
11	8, 9	subge0d 8814	. . . . 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 2626	. 2 ⊢ (∀𝑧 ∈ ℝ (𝑧 ≤ 0 ∨ 0 ≤ 𝑧) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
15		breq2 4115	. . . . 5 ⊢ (𝑦 = 0 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 0))
16		breq1 4114	. . . . 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 4114	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
19		breq2 4115	. . . . . . 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 2544	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧)))
22	21	rspccva 2922	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → ∀𝑦 ∈ ℝ (𝑧 ≤ 𝑦 ∨ 𝑦 ≤ 𝑧))
23		0red 8280	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
24	17, 22, 23	rspcdva 2928	. . 3 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℝ) → (𝑧 ≤ 0 ∨ 0 ≤ 𝑧))
25	24	ralrimiva 2617	. 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 2205  ∀wral 2522   class class class wbr 4111  (class class class)co 6052  ℝcr 8131  0cc0 8132   ≤ cle 8314   − cmin 8449

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-distr 8236  ax-i2m1 8237  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltadd 8248

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452

This theorem is referenced by:  ivthdich  15567