Theorem reapval 8331

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8343 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
reapval	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Proof of Theorem reapval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 3929	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 < 𝑦 ↔ 𝐴 < 𝐵))
2		simpr 109	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3		simpl 108	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
4	2, 3	breq12d 3937	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 < 𝑥 ↔ 𝐵 < 𝐴))
5	1, 4	orbi12d 782	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
6		df-reap 8330	. . 3 ⊢ #ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))}
7	5, 6	brab2ga 4609	. 2 ⊢ (𝐴 #ℝ 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
8	7	baib 904	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480   class class class wbr 3924  ℝcr 7612   < clt 7793   #_ℝ creap 8329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-reap 8330

This theorem is referenced by:  reapirr  8332  recexre  8333  reapti  8334  reaplt  8343