Theorem ltrelpr 6603

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 6568	. 2 ⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))}
2		opabssxp 4414	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝑥) ∧ 𝑞 ∈ (1st ‘𝑦)))} ⊆ (P × P)
3	1, 2	eqsstri 2975	1 ⊢ <P ⊆ (P × P)

Syntax hints:   ∧ wa 97   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  {copab 3817   × cxp 4343  ‘cfv 4902  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378  Pcnp 6389  <_P cltp 6393

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351  df-iltp 6568

This theorem is referenced by:  ltprordil  6687  ltexprlemm  6698  ltexprlemopl  6699  ltexprlemlol  6700  ltexprlemopu  6701  ltexprlemupu  6702  ltexprlemdisj  6704  ltexprlemloc  6705  ltexprlemfl  6707  ltexprlemrl  6708  ltexprlemfu  6709  ltexprlemru  6710  ltexpri  6711  lteupri  6715  ltaprlem  6716  prplnqu  6718  caucvgprprlemk  6781  caucvgprprlemnkltj  6787  caucvgprprlemnkeqj  6788  caucvgprprlemnjltk  6789  caucvgprprlemnbj  6791  caucvgprprlemml  6792  caucvgprprlemlol  6796  caucvgprprlemupu  6798  gt0srpr  6833  lttrsr  6847  ltposr  6848  archsr  6866