Metamath Proof Explorer


Theorem sn-sup2

Description: sup2 with exactly the same proof except for using sn-ltp1 instead of ltp1 , saving ax-mulcom . (Contributed by SN, 26-Jun-2024)

Ref Expression
Assertion sn-sup2 ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 peano2re ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ )
2 1 adantr ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( 𝑥 + 1 ) ∈ ℝ )
3 2 a1i ( 𝐴 ⊆ ℝ → ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( 𝑥 + 1 ) ∈ ℝ ) )
4 ssel ( 𝐴 ⊆ ℝ → ( 𝑦𝐴𝑦 ∈ ℝ ) )
5 sn-ltp1 ( 𝑥 ∈ ℝ → 𝑥 < ( 𝑥 + 1 ) )
6 1 ancli ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ℝ ∧ ( 𝑥 + 1 ) ∈ ℝ ) )
7 lttr ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ( 𝑥 + 1 ) ∈ ℝ ) → ( ( 𝑦 < 𝑥𝑥 < ( 𝑥 + 1 ) ) → 𝑦 < ( 𝑥 + 1 ) ) )
8 7 3expb ( ( 𝑦 ∈ ℝ ∧ ( 𝑥 ∈ ℝ ∧ ( 𝑥 + 1 ) ∈ ℝ ) ) → ( ( 𝑦 < 𝑥𝑥 < ( 𝑥 + 1 ) ) → 𝑦 < ( 𝑥 + 1 ) ) )
9 6 8 sylan2 ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦 < 𝑥𝑥 < ( 𝑥 + 1 ) ) → 𝑦 < ( 𝑥 + 1 ) ) )
10 5 9 sylan2i ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦 < 𝑥𝑥 ∈ ℝ ) → 𝑦 < ( 𝑥 + 1 ) ) )
11 10 exp4b ( 𝑦 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑦 < 𝑥 → ( 𝑥 ∈ ℝ → 𝑦 < ( 𝑥 + 1 ) ) ) ) )
12 11 com34 ( 𝑦 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑦 < 𝑥𝑦 < ( 𝑥 + 1 ) ) ) ) )
13 12 pm2.43d ( 𝑦 ∈ ℝ → ( 𝑥 ∈ ℝ → ( 𝑦 < 𝑥𝑦 < ( 𝑥 + 1 ) ) ) )
14 13 imp ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑦 < 𝑥𝑦 < ( 𝑥 + 1 ) ) )
15 breq1 ( 𝑦 = 𝑥 → ( 𝑦 < ( 𝑥 + 1 ) ↔ 𝑥 < ( 𝑥 + 1 ) ) )
16 5 15 syl5ibrcom ( 𝑥 ∈ ℝ → ( 𝑦 = 𝑥𝑦 < ( 𝑥 + 1 ) ) )
17 16 adantl ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑦 = 𝑥𝑦 < ( 𝑥 + 1 ) ) )
18 14 17 jaod ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦 < 𝑥𝑦 = 𝑥 ) → 𝑦 < ( 𝑥 + 1 ) ) )
19 18 ex ( 𝑦 ∈ ℝ → ( 𝑥 ∈ ℝ → ( ( 𝑦 < 𝑥𝑦 = 𝑥 ) → 𝑦 < ( 𝑥 + 1 ) ) ) )
20 4 19 syl6 ( 𝐴 ⊆ ℝ → ( 𝑦𝐴 → ( 𝑥 ∈ ℝ → ( ( 𝑦 < 𝑥𝑦 = 𝑥 ) → 𝑦 < ( 𝑥 + 1 ) ) ) ) )
21 20 com23 ( 𝐴 ⊆ ℝ → ( 𝑥 ∈ ℝ → ( 𝑦𝐴 → ( ( 𝑦 < 𝑥𝑦 = 𝑥 ) → 𝑦 < ( 𝑥 + 1 ) ) ) ) )
22 21 imp ( ( 𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑦𝐴 → ( ( 𝑦 < 𝑥𝑦 = 𝑥 ) → 𝑦 < ( 𝑥 + 1 ) ) ) )
23 22 a2d ( ( 𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( ( 𝑦𝐴 → ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( 𝑦𝐴𝑦 < ( 𝑥 + 1 ) ) ) )
24 23 ralimdv2 ( ( 𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) → ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) )
25 24 expimpd ( 𝐴 ⊆ ℝ → ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) )
26 3 25 jcad ( 𝐴 ⊆ ℝ → ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) ) )
27 ovex ( 𝑥 + 1 ) ∈ V
28 eleq1 ( 𝑧 = ( 𝑥 + 1 ) → ( 𝑧 ∈ ℝ ↔ ( 𝑥 + 1 ) ∈ ℝ ) )
29 breq2 ( 𝑧 = ( 𝑥 + 1 ) → ( 𝑦 < 𝑧𝑦 < ( 𝑥 + 1 ) ) )
30 29 ralbidv ( 𝑧 = ( 𝑥 + 1 ) → ( ∀ 𝑦𝐴 𝑦 < 𝑧 ↔ ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) )
31 28 30 anbi12d ( 𝑧 = ( 𝑥 + 1 ) → ( ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) ↔ ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) ) )
32 27 31 spcev ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < ( 𝑥 + 1 ) ) → ∃ 𝑧 ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) )
33 26 32 syl6 ( 𝐴 ⊆ ℝ → ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∃ 𝑧 ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) ) )
34 33 exlimdv ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∃ 𝑧 ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) ) )
35 eleq1 ( 𝑧 = 𝑥 → ( 𝑧 ∈ ℝ ↔ 𝑥 ∈ ℝ ) )
36 breq2 ( 𝑧 = 𝑥 → ( 𝑦 < 𝑧𝑦 < 𝑥 ) )
37 36 ralbidv ( 𝑧 = 𝑥 → ( ∀ 𝑦𝐴 𝑦 < 𝑧 ↔ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
38 35 37 anbi12d ( 𝑧 = 𝑥 → ( ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑥 ) ) )
39 38 cbvexvw ( ∃ 𝑧 ( 𝑧 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑧 ) ↔ ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
40 34 39 syl6ib ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑥 ) ) )
41 df-rex ( ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) )
42 df-rex ( ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ ℝ ∧ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
43 40 41 42 3imtr4g ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
44 43 adantr ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
45 44 imdistani ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
46 df-3an ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) ↔ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) )
47 df-3an ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) ↔ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ) ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
48 45 46 47 3imtr4i ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) )
49 axsup ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )
50 48 49 syl ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 ( 𝑦 < 𝑥𝑦 = 𝑥 ) ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )