Theorem ltrelpr 6601

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 6566	. 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 6376  Pcnp 6387  <_P cltp 6391

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 6566

This theorem is referenced by:  ltprordil  6685  ltexprlemm  6696  ltexprlemopl  6697  ltexprlemlol  6698  ltexprlemopu  6699  ltexprlemupu  6700  ltexprlemdisj  6702  ltexprlemloc  6703  ltexprlemfl  6705  ltexprlemrl  6706  ltexprlemfu  6707  ltexprlemru  6708  ltexpri  6709  lteupri  6713  ltaprlem  6714  prplnqu  6716  caucvgprprlemk  6779  caucvgprprlemnkltj  6785  caucvgprprlemnkeqj  6786  caucvgprprlemnjltk  6787  caucvgprprlemnbj  6789  caucvgprprlemml  6790  caucvgprprlemlol  6794  caucvgprprlemupu  6796  gt0srpr  6831  lttrsr  6845  ltposr  6846  archsr  6864