Theorem reapval 8574

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8586 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 4030	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 < 𝑦 ↔ 𝐴 < 𝐵))
2		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3		simpl 109	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
4	2, 3	breq12d 4038	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 < 𝑥 ↔ 𝐵 < 𝐴))
5	1, 4	orbi12d 794	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
6		df-reap 8573	. . 3 ⊢ #ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))}
7	5, 6	brab2ga 4726	. 2 ⊢ (𝐴 #ℝ 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
8	7	baib 920	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160   class class class wbr 4025  ℝcr 7850   < clt 8033   #_ℝ creap 8572

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-br 4026  df-opab 4087  df-xp 4657  df-reap 8573

This theorem is referenced by:  reapirr  8575  recexre  8576  reapti  8577  reaplt  8586