Theorem reapval 8656

Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8668 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 4052	. . . 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 4060	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 < 𝑥 ↔ 𝐵 < 𝐴))
5	1, 4	orbi12d 795	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
6		df-reap 8655	. . 3 ⊢ #ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))}
7	5, 6	brab2ga 4754	. 2 ⊢ (𝐴 #ℝ 𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
8	7	baib 921	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2177   class class class wbr 4047  ℝcr 7931   < clt 8114   #_ℝ creap 8654

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-reap 8655

This theorem is referenced by:  reapirr  8657  recexre  8658  reapti  8659  reaplt  8668