Theorem ltrelpr 6488

Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)

Assertion

Ref	Expression
ltrelpr	⊢ <P ⊆ (P × P)

Proof of Theorem ltrelpr

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

Step	Hyp	Ref	Expression
1		df-iltp 6453	. 2 ⊢ <P = {⟨x, y⟩ ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))}
2		opabssxp 4357	. 2 ⊢ {⟨x, y⟩ ∣ ((x ∈ P ∧ y ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘x) ∧ 𝑞 ∈ (1st ‘y)))} ⊆ (P × P)
3	1, 2	eqsstri 2969	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 97   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  {copab 3808   × cxp 4286  ‘cfv 4845  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264  Pcnp 6275  <_P cltp 6279

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810  df-xp 4294  df-iltp 6453

This theorem is referenced by:  ltprordil  6565  ltexprlemm  6574  ltexprlemopl  6575  ltexprlemlol  6576  ltexprlemopu  6577  ltexprlemupu  6578  ltexprlemdisj  6580  ltexprlemloc  6581  ltexprlemfl  6583  ltexprlemrl  6584  ltexprlemfu  6585  ltexprlemru  6586  ltexpri  6587  ltaprlem  6591  gt0srpr  6676  lttrsr  6690  ltposr  6691  archsr  6708