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Theorem axpre-sup 11112
Description: A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 11237. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11136. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-sup ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
Distinct variable group:   π‘₯,𝑦,𝑧,𝐴

Proof of Theorem axpre-sup
Dummy variables 𝑀 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal2 11075 . . . . . . 7 (π‘₯ ∈ ℝ ↔ ((1st β€˜π‘₯) ∈ R ∧ π‘₯ = ⟨(1st β€˜π‘₯), 0R⟩))
21simplbi 499 . . . . . 6 (π‘₯ ∈ ℝ β†’ (1st β€˜π‘₯) ∈ R)
32adantl 483 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ π‘₯ ∈ ℝ) β†’ (1st β€˜π‘₯) ∈ R)
4 fo1st 7946 . . . . . . . . . . . 12 1st :V–ontoβ†’V
5 fof 6761 . . . . . . . . . . . 12 (1st :V–ontoβ†’V β†’ 1st :V⟢V)
6 ffn 6673 . . . . . . . . . . . 12 (1st :V⟢V β†’ 1st Fn V)
74, 5, 6mp2b 10 . . . . . . . . . . 11 1st Fn V
8 ssv 3973 . . . . . . . . . . 11 𝐴 βŠ† V
9 fvelimab 6919 . . . . . . . . . . 11 ((1st Fn V ∧ 𝐴 βŠ† V) β†’ (𝑀 ∈ (1st β€œ 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐴 (1st β€˜π‘¦) = 𝑀))
107, 8, 9mp2an 691 . . . . . . . . . 10 (𝑀 ∈ (1st β€œ 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐴 (1st β€˜π‘¦) = 𝑀)
11 r19.29 3118 . . . . . . . . . . . 12 ((βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐴 (1st β€˜π‘¦) = 𝑀) β†’ βˆƒπ‘¦ ∈ 𝐴 (𝑦 <ℝ π‘₯ ∧ (1st β€˜π‘¦) = 𝑀))
12 ssel2 3944 . . . . . . . . . . . . . . . . 17 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ ℝ)
13 ltresr2 11084 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ ∧ π‘₯ ∈ ℝ) β†’ (𝑦 <ℝ π‘₯ ↔ (1st β€˜π‘¦) <R (1st β€˜π‘₯)))
14 breq1 5113 . . . . . . . . . . . . . . . . . . . 20 ((1st β€˜π‘¦) = 𝑀 β†’ ((1st β€˜π‘¦) <R (1st β€˜π‘₯) ↔ 𝑀 <R (1st β€˜π‘₯)))
1513, 14sylan9bb 511 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ ℝ ∧ π‘₯ ∈ ℝ) ∧ (1st β€˜π‘¦) = 𝑀) β†’ (𝑦 <ℝ π‘₯ ↔ 𝑀 <R (1st β€˜π‘₯)))
1615biimpd 228 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℝ ∧ π‘₯ ∈ ℝ) ∧ (1st β€˜π‘¦) = 𝑀) β†’ (𝑦 <ℝ π‘₯ β†’ 𝑀 <R (1st β€˜π‘₯)))
1716exp31 421 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ β†’ (π‘₯ ∈ ℝ β†’ ((1st β€˜π‘¦) = 𝑀 β†’ (𝑦 <ℝ π‘₯ β†’ 𝑀 <R (1st β€˜π‘₯)))))
1812, 17syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ ℝ β†’ ((1st β€˜π‘¦) = 𝑀 β†’ (𝑦 <ℝ π‘₯ β†’ 𝑀 <R (1st β€˜π‘₯)))))
1918imp4b 423 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) ∧ π‘₯ ∈ ℝ) β†’ (((1st β€˜π‘¦) = 𝑀 ∧ 𝑦 <ℝ π‘₯) β†’ 𝑀 <R (1st β€˜π‘₯)))
2019ancomsd 467 . . . . . . . . . . . . . 14 (((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) ∧ π‘₯ ∈ ℝ) β†’ ((𝑦 <ℝ π‘₯ ∧ (1st β€˜π‘¦) = 𝑀) β†’ 𝑀 <R (1st β€˜π‘₯)))
2120an32s 651 . . . . . . . . . . . . 13 (((𝐴 βŠ† ℝ ∧ π‘₯ ∈ ℝ) ∧ 𝑦 ∈ 𝐴) β†’ ((𝑦 <ℝ π‘₯ ∧ (1st β€˜π‘¦) = 𝑀) β†’ 𝑀 <R (1st β€˜π‘₯)))
2221rexlimdva 3153 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ π‘₯ ∈ ℝ) β†’ (βˆƒπ‘¦ ∈ 𝐴 (𝑦 <ℝ π‘₯ ∧ (1st β€˜π‘¦) = 𝑀) β†’ 𝑀 <R (1st β€˜π‘₯)))
2311, 22syl5 34 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ π‘₯ ∈ ℝ) β†’ ((βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ ∧ βˆƒπ‘¦ ∈ 𝐴 (1st β€˜π‘¦) = 𝑀) β†’ 𝑀 <R (1st β€˜π‘₯)))
2423expd 417 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ (βˆƒπ‘¦ ∈ 𝐴 (1st β€˜π‘¦) = 𝑀 β†’ 𝑀 <R (1st β€˜π‘₯))))
2510, 24syl7bi 255 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ (𝑀 ∈ (1st β€œ 𝐴) β†’ 𝑀 <R (1st β€˜π‘₯))))
2625impr 456 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ (π‘₯ ∈ ℝ ∧ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯)) β†’ (𝑀 ∈ (1st β€œ 𝐴) β†’ 𝑀 <R (1st β€˜π‘₯)))
2726adantlr 714 . . . . . . 7 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ (π‘₯ ∈ ℝ ∧ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯)) β†’ (𝑀 ∈ (1st β€œ 𝐴) β†’ 𝑀 <R (1st β€˜π‘₯)))
2827ralrimiv 3143 . . . . . 6 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ (π‘₯ ∈ ℝ ∧ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯)) β†’ βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R (1st β€˜π‘₯))
2928expr 458 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R (1st β€˜π‘₯)))
30 brralrspcev 5170 . . . . 5 (((1st β€˜π‘₯) ∈ R ∧ βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R (1st β€˜π‘₯)) β†’ βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣)
313, 29, 30syl6an 683 . . . 4 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣))
3231rexlimdva 3153 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣))
33 n0 4311 . . . . . 6 (𝐴 β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
34 fnfvima 7188 . . . . . . . . 9 ((1st Fn V ∧ 𝐴 βŠ† V ∧ 𝑦 ∈ 𝐴) β†’ (1st β€˜π‘¦) ∈ (1st β€œ 𝐴))
357, 8, 34mp3an12 1452 . . . . . . . 8 (𝑦 ∈ 𝐴 β†’ (1st β€˜π‘¦) ∈ (1st β€œ 𝐴))
3635ne0d 4300 . . . . . . 7 (𝑦 ∈ 𝐴 β†’ (1st β€œ 𝐴) β‰  βˆ…)
3736exlimiv 1934 . . . . . 6 (βˆƒπ‘¦ 𝑦 ∈ 𝐴 β†’ (1st β€œ 𝐴) β‰  βˆ…)
3833, 37sylbi 216 . . . . 5 (𝐴 β‰  βˆ… β†’ (1st β€œ 𝐴) β‰  βˆ…)
39 supsr 11055 . . . . . 6 (((1st β€œ 𝐴) β‰  βˆ… ∧ βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣) β†’ βˆƒπ‘£ ∈ R (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒)))
4039ex 414 . . . . 5 ((1st β€œ 𝐴) β‰  βˆ… β†’ (βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣 β†’ βˆƒπ‘£ ∈ R (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒))))
4138, 40syl 17 . . . 4 (𝐴 β‰  βˆ… β†’ (βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣 β†’ βˆƒπ‘£ ∈ R (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒))))
4241adantl 483 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (βˆƒπ‘£ ∈ R βˆ€π‘€ ∈ (1st β€œ 𝐴)𝑀 <R 𝑣 β†’ βˆƒπ‘£ ∈ R (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒))))
43 breq2 5114 . . . . . . . . . . . 12 (𝑀 = (1st β€˜π‘¦) β†’ (𝑣 <R 𝑀 ↔ 𝑣 <R (1st β€˜π‘¦)))
4443notbid 318 . . . . . . . . . . 11 (𝑀 = (1st β€˜π‘¦) β†’ (Β¬ 𝑣 <R 𝑀 ↔ Β¬ 𝑣 <R (1st β€˜π‘¦)))
4544rspccv 3581 . . . . . . . . . 10 (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ ((1st β€˜π‘¦) ∈ (1st β€œ 𝐴) β†’ Β¬ 𝑣 <R (1st β€˜π‘¦)))
4635, 45syl5com 31 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ Β¬ 𝑣 <R (1st β€˜π‘¦)))
4746adantl 483 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ Β¬ 𝑣 <R (1st β€˜π‘¦)))
48 elreal2 11075 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ ↔ ((1st β€˜π‘¦) ∈ R ∧ 𝑦 = ⟨(1st β€˜π‘¦), 0R⟩))
4948simprbi 498 . . . . . . . . . . . 12 (𝑦 ∈ ℝ β†’ 𝑦 = ⟨(1st β€˜π‘¦), 0R⟩)
5049breq2d 5122 . . . . . . . . . . 11 (𝑦 ∈ ℝ β†’ (βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ↔ βŸ¨π‘£, 0R⟩ <ℝ ⟨(1st β€˜π‘¦), 0R⟩))
51 ltresr 11083 . . . . . . . . . . 11 (βŸ¨π‘£, 0R⟩ <ℝ ⟨(1st β€˜π‘¦), 0R⟩ ↔ 𝑣 <R (1st β€˜π‘¦))
5250, 51bitrdi 287 . . . . . . . . . 10 (𝑦 ∈ ℝ β†’ (βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st β€˜π‘¦)))
5312, 52syl 17 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ (βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st β€˜π‘¦)))
5453notbid 318 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ (Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ↔ Β¬ 𝑣 <R (1st β€˜π‘¦)))
5547, 54sylibrd 259 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ 𝑦 ∈ 𝐴) β†’ (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
5655ralrimdva 3152 . . . . . 6 (𝐴 βŠ† ℝ β†’ (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
5756ad2antrr 725 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 β†’ βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
5849breq1d 5120 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ ↔ ⟨(1st β€˜π‘¦), 0R⟩ <ℝ βŸ¨π‘£, 0R⟩))
59 ltresr 11083 . . . . . . . . . . . . . 14 (⟨(1st β€˜π‘¦), 0R⟩ <ℝ βŸ¨π‘£, 0R⟩ ↔ (1st β€˜π‘¦) <R 𝑣)
6058, 59bitrdi 287 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ ↔ (1st β€˜π‘¦) <R 𝑣))
6148simplbi 499 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ β†’ (1st β€˜π‘¦) ∈ R)
62 breq1 5113 . . . . . . . . . . . . . . . . 17 (𝑀 = (1st β€˜π‘¦) β†’ (𝑀 <R 𝑣 ↔ (1st β€˜π‘¦) <R 𝑣))
63 breq1 5113 . . . . . . . . . . . . . . . . . 18 (𝑀 = (1st β€˜π‘¦) β†’ (𝑀 <R 𝑒 ↔ (1st β€˜π‘¦) <R 𝑒))
6463rexbidv 3176 . . . . . . . . . . . . . . . . 17 (𝑀 = (1st β€˜π‘¦) β†’ (βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒 ↔ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒))
6562, 64imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑀 = (1st β€˜π‘¦) β†’ ((𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) ↔ ((1st β€˜π‘¦) <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
6665rspccv 3581 . . . . . . . . . . . . . . 15 (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ ((1st β€˜π‘¦) ∈ R β†’ ((1st β€˜π‘¦) <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
6761, 66syl5 34 . . . . . . . . . . . . . 14 (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ (𝑦 ∈ ℝ β†’ ((1st β€˜π‘¦) <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
6867com3l 89 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ β†’ ((1st β€˜π‘¦) <R 𝑣 β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
6960, 68sylbid 239 . . . . . . . . . . . 12 (𝑦 ∈ ℝ β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
7069adantr 482 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒)))
71 fvelimab 6919 . . . . . . . . . . . . . . . 16 ((1st Fn V ∧ 𝐴 βŠ† V) β†’ (𝑒 ∈ (1st β€œ 𝐴) ↔ βˆƒπ‘§ ∈ 𝐴 (1st β€˜π‘§) = 𝑒))
727, 8, 71mp2an 691 . . . . . . . . . . . . . . 15 (𝑒 ∈ (1st β€œ 𝐴) ↔ βˆƒπ‘§ ∈ 𝐴 (1st β€˜π‘§) = 𝑒)
73 ssel2 3944 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 ∈ ℝ)
74 ltresr2 11084 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ (𝑦 <ℝ 𝑧 ↔ (1st β€˜π‘¦) <R (1st β€˜π‘§)))
7573, 74sylan2 594 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴)) β†’ (𝑦 <ℝ 𝑧 ↔ (1st β€˜π‘¦) <R (1st β€˜π‘§)))
76 breq2 5114 . . . . . . . . . . . . . . . . . . . . 21 ((1st β€˜π‘§) = 𝑒 β†’ ((1st β€˜π‘¦) <R (1st β€˜π‘§) ↔ (1st β€˜π‘¦) <R 𝑒))
7775, 76sylan9bb 511 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℝ ∧ (𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴)) ∧ (1st β€˜π‘§) = 𝑒) β†’ (𝑦 <ℝ 𝑧 ↔ (1st β€˜π‘¦) <R 𝑒))
7877exbiri 810 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ ∧ (𝐴 βŠ† ℝ ∧ 𝑧 ∈ 𝐴)) β†’ ((1st β€˜π‘§) = 𝑒 β†’ ((1st β€˜π‘¦) <R 𝑒 β†’ 𝑦 <ℝ 𝑧)))
7978expr 458 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (𝑧 ∈ 𝐴 β†’ ((1st β€˜π‘§) = 𝑒 β†’ ((1st β€˜π‘¦) <R 𝑒 β†’ 𝑦 <ℝ 𝑧))))
8079com4r 94 . . . . . . . . . . . . . . . . 17 ((1st β€˜π‘¦) <R 𝑒 β†’ ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (𝑧 ∈ 𝐴 β†’ ((1st β€˜π‘§) = 𝑒 β†’ 𝑦 <ℝ 𝑧))))
8180imp 408 . . . . . . . . . . . . . . . 16 (((1st β€˜π‘¦) <R 𝑒 ∧ (𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ)) β†’ (𝑧 ∈ 𝐴 β†’ ((1st β€˜π‘§) = 𝑒 β†’ 𝑦 <ℝ 𝑧)))
8281reximdvai 3163 . . . . . . . . . . . . . . 15 (((1st β€˜π‘¦) <R 𝑒 ∧ (𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ)) β†’ (βˆƒπ‘§ ∈ 𝐴 (1st β€˜π‘§) = 𝑒 β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))
8372, 82biimtrid 241 . . . . . . . . . . . . . 14 (((1st β€˜π‘¦) <R 𝑒 ∧ (𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ)) β†’ (𝑒 ∈ (1st β€œ 𝐴) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))
8483expcom 415 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ ((1st β€˜π‘¦) <R 𝑒 β†’ (𝑒 ∈ (1st β€œ 𝐴) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
8584com23 86 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (𝑒 ∈ (1st β€œ 𝐴) β†’ ((1st β€˜π‘¦) <R 𝑒 β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
8685rexlimdv 3151 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (βˆƒπ‘’ ∈ (1st β€œ 𝐴)(1st β€˜π‘¦) <R 𝑒 β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))
8770, 86syl6d 75 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
8887com23 86 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝐴 βŠ† ℝ) β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
8988ex 414 . . . . . . . 8 (𝑦 ∈ ℝ β†’ (𝐴 βŠ† ℝ β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
9089com3l 89 . . . . . . 7 (𝐴 βŠ† ℝ β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ (𝑦 ∈ ℝ β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
9190ad2antrr 725 . . . . . 6 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ (𝑦 ∈ ℝ β†’ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
9291ralrimdv 3150 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ (βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒) β†’ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
93 opelreal 11073 . . . . . . . 8 (βŸ¨π‘£, 0R⟩ ∈ ℝ ↔ 𝑣 ∈ R)
9493biimpri 227 . . . . . . 7 (𝑣 ∈ R β†’ βŸ¨π‘£, 0R⟩ ∈ ℝ)
9594adantl 483 . . . . . 6 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ βŸ¨π‘£, 0R⟩ ∈ ℝ)
96 breq1 5113 . . . . . . . . . . 11 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ (π‘₯ <ℝ 𝑦 ↔ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
9796notbid 318 . . . . . . . . . 10 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ (Β¬ π‘₯ <ℝ 𝑦 ↔ Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
9897ralbidv 3175 . . . . . . . . 9 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ↔ βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦))
99 breq2 5114 . . . . . . . . . . 11 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ (𝑦 <ℝ π‘₯ ↔ 𝑦 <ℝ βŸ¨π‘£, 0R⟩))
10099imbi1d 342 . . . . . . . . . 10 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ ((𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
101100ralbidv 3175 . . . . . . . . 9 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ (βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
10298, 101anbi12d 632 . . . . . . . 8 (π‘₯ = βŸ¨π‘£, 0R⟩ β†’ ((βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)) ↔ (βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
103102rspcev 3584 . . . . . . 7 ((βŸ¨π‘£, 0R⟩ ∈ ℝ ∧ (βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
104103ex 414 . . . . . 6 (βŸ¨π‘£, 0R⟩ ∈ ℝ β†’ ((βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
10595, 104syl 17 . . . . 5 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ ((βˆ€π‘¦ ∈ 𝐴 Β¬ βŸ¨π‘£, 0R⟩ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ βŸ¨π‘£, 0R⟩ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
10657, 92, 105syl2and 609 . . . 4 (((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) ∧ 𝑣 ∈ R) β†’ ((βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒)) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
107106rexlimdva 3153 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (βˆƒπ‘£ ∈ R (βˆ€π‘€ ∈ (1st β€œ 𝐴) Β¬ 𝑣 <R 𝑀 ∧ βˆ€π‘€ ∈ R (𝑀 <R 𝑣 β†’ βˆƒπ‘’ ∈ (1st β€œ 𝐴)𝑀 <R 𝑒)) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
10832, 42, 1073syld 60 . 2 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧))))
1091083impia 1118 1 ((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 <ℝ π‘₯) β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ <ℝ 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 <ℝ π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 <ℝ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  βŸ¨cop 4597   class class class wbr 5110   β€œ cima 5641   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€˜cfv 6501  1st c1st 7924  Rcnr 10808  0Rc0r 10809   <R cltr 10814  β„cr 11057   <ℝ cltrr 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-ec 8657  df-qs 8661  df-ni 10815  df-pli 10816  df-mi 10817  df-lti 10818  df-plpq 10851  df-mpq 10852  df-ltpq 10853  df-enq 10854  df-nq 10855  df-erq 10856  df-plq 10857  df-mq 10858  df-1nq 10859  df-rq 10860  df-ltnq 10861  df-np 10924  df-1p 10925  df-plp 10926  df-mp 10927  df-ltp 10928  df-enr 10998  df-nr 10999  df-plr 11000  df-mr 11001  df-ltr 11002  df-0r 11003  df-1r 11004  df-m1r 11005  df-r 11068  df-lt 11071
This theorem is referenced by: (None)
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