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Theorem suplem2pr 10038
 Description: The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
suplem2pr (𝐴P → ((𝑦𝐴 → ¬ 𝐴<P 𝑦) ∧ (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧)))
Distinct variable group:   𝑦,𝑧,𝐴

Proof of Theorem suplem2pr
StepHypRef Expression
1 ltrelpr 9983 . . . . . 6 <P ⊆ (P × P)
21brel 5313 . . . . 5 (𝑦<P 𝐴 → (𝑦P 𝐴P))
32simpld 477 . . . 4 (𝑦<P 𝐴𝑦P)
4 ralnex 3118 . . . . . . . . 9 (∀𝑧𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧𝐴 𝑦<P 𝑧)
5 ssel2 3727 . . . . . . . . . . . 12 ((𝐴P𝑧𝐴) → 𝑧P)
6 ltsopr 10017 . . . . . . . . . . . . . . . 16 <P Or P
7 sotric 5201 . . . . . . . . . . . . . . . 16 ((<P Or P ∧ (𝑦P𝑧P)) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧𝑧<P 𝑦)))
86, 7mpan 708 . . . . . . . . . . . . . . 15 ((𝑦P𝑧P) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧𝑧<P 𝑦)))
98con2bid 343 . . . . . . . . . . . . . 14 ((𝑦P𝑧P) → ((𝑦 = 𝑧𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
109ancoms 468 . . . . . . . . . . . . 13 ((𝑧P𝑦P) → ((𝑦 = 𝑧𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11 ltprord 10015 . . . . . . . . . . . . . . 15 ((𝑧P𝑦P) → (𝑧<P 𝑦𝑧𝑦))
1211orbi2d 740 . . . . . . . . . . . . . 14 ((𝑧P𝑦P) → ((𝑦 = 𝑧𝑧<P 𝑦) ↔ (𝑦 = 𝑧𝑧𝑦)))
13 sspss 3836 . . . . . . . . . . . . . . 15 (𝑧𝑦 ↔ (𝑧𝑦𝑧 = 𝑦))
14 equcom 2088 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦𝑦 = 𝑧)
1514orbi2i 542 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑧 = 𝑦) ↔ (𝑧𝑦𝑦 = 𝑧))
16 orcom 401 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦 = 𝑧) ↔ (𝑦 = 𝑧𝑧𝑦))
1713, 15, 163bitri 286 . . . . . . . . . . . . . 14 (𝑧𝑦 ↔ (𝑦 = 𝑧𝑧𝑦))
1812, 17syl6bbr 278 . . . . . . . . . . . . 13 ((𝑧P𝑦P) → ((𝑦 = 𝑧𝑧<P 𝑦) ↔ 𝑧𝑦))
1910, 18bitr3d 270 . . . . . . . . . . . 12 ((𝑧P𝑦P) → (¬ 𝑦<P 𝑧𝑧𝑦))
205, 19sylan 489 . . . . . . . . . . 11 (((𝐴P𝑧𝐴) ∧ 𝑦P) → (¬ 𝑦<P 𝑧𝑧𝑦))
2120an32s 881 . . . . . . . . . 10 (((𝐴P𝑦P) ∧ 𝑧𝐴) → (¬ 𝑦<P 𝑧𝑧𝑦))
2221ralbidva 3111 . . . . . . . . 9 ((𝐴P𝑦P) → (∀𝑧𝐴 ¬ 𝑦<P 𝑧 ↔ ∀𝑧𝐴 𝑧𝑦))
234, 22syl5bbr 274 . . . . . . . 8 ((𝐴P𝑦P) → (¬ ∃𝑧𝐴 𝑦<P 𝑧 ↔ ∀𝑧𝐴 𝑧𝑦))
24 unissb 4609 . . . . . . . 8 ( 𝐴𝑦 ↔ ∀𝑧𝐴 𝑧𝑦)
2523, 24syl6bbr 278 . . . . . . 7 ((𝐴P𝑦P) → (¬ ∃𝑧𝐴 𝑦<P 𝑧 𝐴𝑦))
26 ssnpss 3840 . . . . . . . 8 ( 𝐴𝑦 → ¬ 𝑦 𝐴)
27 ltprord 10015 . . . . . . . . . 10 ((𝑦P 𝐴P) → (𝑦<P 𝐴𝑦 𝐴))
2827biimpd 219 . . . . . . . . 9 ((𝑦P 𝐴P) → (𝑦<P 𝐴𝑦 𝐴))
292, 28mpcom 38 . . . . . . . 8 (𝑦<P 𝐴𝑦 𝐴)
3026, 29nsyl 135 . . . . . . 7 ( 𝐴𝑦 → ¬ 𝑦<P 𝐴)
3125, 30syl6bi 243 . . . . . 6 ((𝐴P𝑦P) → (¬ ∃𝑧𝐴 𝑦<P 𝑧 → ¬ 𝑦<P 𝐴))
3231con4d 114 . . . . 5 ((𝐴P𝑦P) → (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧))
3332ex 449 . . . 4 (𝐴P → (𝑦P → (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧)))
343, 33syl5 34 . . 3 (𝐴P → (𝑦<P 𝐴 → (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧)))
3534pm2.43d 53 . 2 (𝐴P → (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧))
36 elssuni 4607 . . . 4 (𝑦𝐴𝑦 𝐴)
37 ssnpss 3840 . . . 4 (𝑦 𝐴 → ¬ 𝐴𝑦)
3836, 37syl 17 . . 3 (𝑦𝐴 → ¬ 𝐴𝑦)
391brel 5313 . . . 4 ( 𝐴<P 𝑦 → ( 𝐴P𝑦P))
40 ltprord 10015 . . . . 5 (( 𝐴P𝑦P) → ( 𝐴<P 𝑦 𝐴𝑦))
4140biimpd 219 . . . 4 (( 𝐴P𝑦P) → ( 𝐴<P 𝑦 𝐴𝑦))
4239, 41mpcom 38 . . 3 ( 𝐴<P 𝑦 𝐴𝑦)
4338, 42nsyl 135 . 2 (𝑦𝐴 → ¬ 𝐴<P 𝑦)
4435, 43jctil 561 1 (𝐴P → ((𝑦𝐴 → ¬ 𝐴<P 𝑦) ∧ (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∈ wcel 2127  ∀wral 3038  ∃wrex 3039   ⊆ wss 3703   ⊊ wpss 3704  ∪ cuni 4576   class class class wbr 4792   Or wor 5174  Pcnp 9844
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