Theorem ltprord 11115

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 2848	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ P ↔ 𝐴 ∈ P))
2	1	anbi1d 643	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝑦 ∈ P)))
3		psseq1 4017	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝑦))
4	2, 3	anbi12d 644	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦)))
5		eleq1 2848	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ P ↔ 𝐵 ∈ P))
6	5	anbi2d 642	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ P) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ P)))
7		psseq2 4018	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ⊊ 𝑦 ↔ 𝐴 ⊊ 𝐵))
8	6, 7	anbi12d 644	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ P) ∧ 𝐴 ⊊ 𝑦) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵)))
9		df-ltp 11070	. . 3 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
10	4, 8, 9	brabg 5513	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵)))
11	10	bianabs 551	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊊ wpss 4013   class class class wbr 5102  Pcnp 10944  <_P cltp 10948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-ltp 11070

This theorem is used by:  ltsopr  11117  ltaddpr  11119  ltexprlem7  11127  ltexpri  11128  suplem1pr  11137  suplem2pr  11138