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Theorem chnso 33005
Description: A chain induces a total order. (Contributed by Thierry Arnoux, 19-Jun-2025.)
Assertion
Ref Expression
chnso (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)

Proof of Theorem chnso
Dummy variables 𝑥 𝑦 𝑖 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2737 . . . . 5 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → (♯‘𝐶) = (♯‘𝐶))
2 ischn 32997 . . . . . . . 8 (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶𝑛)))
32biimpi 216 . . . . . . 7 (𝐶 ∈ ( < Chain𝐴) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶𝑛)))
43adantl 481 . . . . . 6 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶𝑛)))
54simpld 494 . . . . 5 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → 𝐶 ∈ Word 𝐴)
61, 5wrdfd 32919 . . . 4 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → 𝐶:(0..^(♯‘𝐶))⟶𝐴)
76frnd 6743 . . 3 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → ran 𝐶𝐴)
8 simpl 482 . . 3 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Po 𝐴)
9 poss 5593 . . 3 (ran 𝐶𝐴 → ( < Po 𝐴< Po ran 𝐶))
107, 8, 9sylc 65 . 2 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Po ran 𝐶)
11 fzossz 13720 . . . . . . . . 9 (0..^(♯‘𝐶)) ⊆ ℤ
12 simp-4r 783 . . . . . . . . 9 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑖 ∈ (0..^(♯‘𝐶)))
1311, 12sselid 3980 . . . . . . . 8 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑖 ∈ ℤ)
1413zred 12724 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑖 ∈ ℝ)
15 simplr 768 . . . . . . . . 9 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑗 ∈ (0..^(♯‘𝐶)))
1611, 15sselid 3980 . . . . . . . 8 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑗 ∈ ℤ)
1716zred 12724 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑗 ∈ ℝ)
1814, 17lttri4d 11403 . . . . . 6 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
19 simp-8l 790 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → < Po 𝐴)
20 simp-8r 791 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝐶 ∈ ( < Chain𝐴))
21 simpllr 775 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0..^(♯‘𝐶)))
22 elfzouz 13704 . . . . . . . . . . . 12 (𝑖 ∈ (0..^(♯‘𝐶)) → 𝑖 ∈ (ℤ‘0))
2322ad5antlr 735 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (ℤ‘0))
2416adantr 480 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
25 simpr 484 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
26 elfzo2 13703 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑗) ↔ (𝑖 ∈ (ℤ‘0) ∧ 𝑗 ∈ ℤ ∧ 𝑖 < 𝑗))
2723, 24, 25, 26syl3anbrc 1343 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0..^𝑗))
2819, 20, 21, 27chnlt 33004 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶𝑖) < (𝐶𝑗))
29 simp-4r 783 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶𝑖) = 𝑥)
30 simplr 768 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶𝑗) = 𝑦)
3128, 29, 303brtr3d 5173 . . . . . . . 8 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑥 < 𝑦)
3231ex 412 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → (𝑖 < 𝑗𝑥 < 𝑦))
33 simpr 484 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
3433fveq2d 6909 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶𝑖) = (𝐶𝑗))
35 simp-4r 783 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶𝑖) = 𝑥)
36 simplr 768 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶𝑗) = 𝑦)
3734, 35, 363eqtr3d 2784 . . . . . . . 8 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → 𝑥 = 𝑦)
3837ex 412 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → (𝑖 = 𝑗𝑥 = 𝑦))
39 simp-8l 790 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → < Po 𝐴)
40 simp-8r 791 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝐶 ∈ ( < Chain𝐴))
4112adantr 480 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0..^(♯‘𝐶)))
42 elfzouz 13704 . . . . . . . . . . . 12 (𝑗 ∈ (0..^(♯‘𝐶)) → 𝑗 ∈ (ℤ‘0))
4342ad3antlr 731 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (ℤ‘0))
4413adantr 480 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑖 ∈ ℤ)
45 simpr 484 . . . . . . . . . . 11 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
46 elfzo2 13703 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑖) ↔ (𝑗 ∈ (ℤ‘0) ∧ 𝑖 ∈ ℤ ∧ 𝑗 < 𝑖))
4743, 44, 45, 46syl3anbrc 1343 . . . . . . . . . 10 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0..^𝑖))
4839, 40, 41, 47chnlt 33004 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶𝑗) < (𝐶𝑖))
49 simplr 768 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶𝑗) = 𝑦)
50 simp-4r 783 . . . . . . . . 9 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶𝑖) = 𝑥)
5148, 49, 503brtr3d 5173 . . . . . . . 8 ((((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑦 < 𝑥)
5251ex 412 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → (𝑗 < 𝑖𝑦 < 𝑥))
5332, 38, 523orim123d 1445 . . . . . 6 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → ((𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
5418, 53mpd 15 . . . . 5 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
556ffnd 6736 . . . . . . 7 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → 𝐶 Fn (0..^(♯‘𝐶)))
5655ad4antr 732 . . . . . 6 (((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) → 𝐶 Fn (0..^(♯‘𝐶)))
57 simpllr 775 . . . . . 6 (((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) → 𝑦 ∈ ran 𝐶)
58 fvelrnb 6968 . . . . . . 7 (𝐶 Fn (0..^(♯‘𝐶)) → (𝑦 ∈ ran 𝐶 ↔ ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶𝑗) = 𝑦))
5958biimpa 476 . . . . . 6 ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶𝑗) = 𝑦)
6056, 57, 59syl2anc 584 . . . . 5 (((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶𝑗) = 𝑦)
6154, 60r19.29a 3161 . . . 4 (((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
6255ad2antrr 726 . . . . 5 (((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝐶 Fn (0..^(♯‘𝐶)))
63 simplr 768 . . . . 5 (((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝑥 ∈ ran 𝐶)
64 fvelrnb 6968 . . . . . 6 (𝐶 Fn (0..^(♯‘𝐶)) → (𝑥 ∈ ran 𝐶 ↔ ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶𝑖) = 𝑥))
6564biimpa 476 . . . . 5 ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑥 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶𝑖) = 𝑥)
6662, 63, 65syl2anc 584 . . . 4 (((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶𝑖) = 𝑥)
6761, 66r19.29a 3161 . . 3 (((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
6867anasss 466 . 2 ((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ (𝑥 ∈ ran 𝐶𝑦 ∈ ran 𝐶)) → (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))
6910, 68issod 5626 1 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085   = wceq 1539  wcel 2107  wral 3060  wrex 3069  cdif 3947  wss 3950  {csn 4625   class class class wbr 5142   Po wpo 5589   Or wor 5590  dom cdm 5684  ran crn 5685   Fn wfn 6555  cfv 6560  (class class class)co 7432  0cc0 11156  1c1 11157   < clt 11296  cmin 11493  cz 12615  cuz 12879  ..^cfzo 13695  chash 14370  Word cword 14553  Chaincchn 32995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-hash 14371  df-word 14554  df-lsw 14602  df-concat 14610  df-s1 14635  df-substr 14680  df-pfx 14710  df-chn 32996
This theorem is referenced by: (None)
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