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Theorem psslinpr 9891
 Description: Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
psslinpr ((𝐴P𝐵P) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Proof of Theorem psslinpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elprnq 9851 . . . . . . . . . . . . 13 ((𝐴P𝑥𝐴) → 𝑥Q)
2 prub 9854 . . . . . . . . . . . . 13 (((𝐵P𝑦𝐵) ∧ 𝑥Q) → (¬ 𝑥𝐵𝑦 <Q 𝑥))
31, 2sylan2 490 . . . . . . . . . . . 12 (((𝐵P𝑦𝐵) ∧ (𝐴P𝑥𝐴)) → (¬ 𝑥𝐵𝑦 <Q 𝑥))
4 prcdnq 9853 . . . . . . . . . . . . 13 ((𝐴P𝑥𝐴) → (𝑦 <Q 𝑥𝑦𝐴))
54adantl 481 . . . . . . . . . . . 12 (((𝐵P𝑦𝐵) ∧ (𝐴P𝑥𝐴)) → (𝑦 <Q 𝑥𝑦𝐴))
63, 5syld 47 . . . . . . . . . . 11 (((𝐵P𝑦𝐵) ∧ (𝐴P𝑥𝐴)) → (¬ 𝑥𝐵𝑦𝐴))
76exp43 639 . . . . . . . . . 10 (𝐵P → (𝑦𝐵 → (𝐴P → (𝑥𝐴 → (¬ 𝑥𝐵𝑦𝐴)))))
87com3r 87 . . . . . . . . 9 (𝐴P → (𝐵P → (𝑦𝐵 → (𝑥𝐴 → (¬ 𝑥𝐵𝑦𝐴)))))
98imp 444 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴 → (¬ 𝑥𝐵𝑦𝐴))))
109imp4a 613 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑦𝐴)))
1110com23 86 . . . . . 6 ((𝐴P𝐵P) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑦𝐵𝑦𝐴)))
1211alrimdv 1897 . . . . 5 ((𝐴P𝐵P) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ∀𝑦(𝑦𝐵𝑦𝐴)))
1312exlimdv 1901 . . . 4 ((𝐴P𝐵P) → (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ∀𝑦(𝑦𝐵𝑦𝐴)))
14 nss 3696 . . . . 5 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
15 sspss 3739 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
1614, 15xchnxbi 321 . . . 4 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
17 sspss 3739 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
18 dfss2 3624 . . . . 5 (𝐵𝐴 ↔ ∀𝑦(𝑦𝐵𝑦𝐴))
1917, 18bitr3i 266 . . . 4 ((𝐵𝐴𝐵 = 𝐴) ↔ ∀𝑦(𝑦𝐵𝑦𝐴))
2013, 16, 193imtr4g 285 . . 3 ((𝐴P𝐵P) → (¬ (𝐴𝐵𝐴 = 𝐵) → (𝐵𝐴𝐵 = 𝐴)))
2120orrd 392 . 2 ((𝐴P𝐵P) → ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐵 = 𝐴)))
22 df-3or 1055 . . 3 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
23 or32 548 . . 3 (((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴) ↔ ((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵))
24 orordir 552 . . . 4 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
25 eqcom 2658 . . . . . 6 (𝐵 = 𝐴𝐴 = 𝐵)
2625orbi2i 540 . . . . 5 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
2726orbi2i 540 . . . 4 (((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐵 = 𝐴)) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐴 = 𝐵)))
2824, 27bitr4i 267 . . 3 (((𝐴𝐵𝐵𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐵 = 𝐴)))
2922, 23, 283bitri 286 . 2 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ (𝐵𝐴𝐵 = 𝐴)))
3021, 29sylibr 224 1 ((𝐴P𝐵P) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1053  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ⊆ wss 3607   ⊊ wpss 3608   class class class wbr 4685  Qcnq 9712
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