Theorem ltprord 10786

Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltprord	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵))

Proof of Theorem ltprord

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

Step	Hyp	Ref	Expression
1		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ P ↔ 𝐴 ∈ P))
2	1	anbi1d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝑦 ∈ P)))
3		psseq1 4022	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦))
4	2, 3	anbi12d 631	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦)))
5		eleq1 2826	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ P ↔ 𝐵 ∈ P))
6	5	anbi2d 629	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P)))
7		psseq2 4023	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵))
8	6, 7	anbi12d 631	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵)))
9		df-ltp 10741	. . 3 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
10	4, 8, 9	brabg 5452	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵)))
11	10	bianabs 542	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊊ wpss 3888   class class class wbr 5074  Pcnp 10615  <_P cltp 10619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-ltp 10741

This theorem is referenced by:  ltsopr  10788  ltaddpr  10790  ltexprlem7  10798  ltexpri  10799  suplem1pr  10808  suplem2pr  10809