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Theorem chnso 32988
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 2736 . . . . 5 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → (♯‘𝐶) = (♯‘𝐶))
2 ischn 32981 . . . . . . . 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 32903 . . . 4 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → 𝐶:(0..^(♯‘𝐶))⟶𝐴)
76frnd 6745 . . 3 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → ran 𝐶𝐴)
8 simpl 482 . . 3 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Po 𝐴)
9 poss 5599 . . 3 (ran 𝐶𝐴 → ( < Po 𝐴< Po ran 𝐶))
107, 8, 9sylc 65 . 2 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Po ran 𝐶)
11 fzossz 13716 . . . . . . . . 9 (0..^(♯‘𝐶)) ⊆ ℤ
12 simp-4r 784 . . . . . . . . 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 3993 . . . . . . . 8 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑖 ∈ ℤ)
1413zred 12720 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑖 ∈ ℝ)
15 simplr 769 . . . . . . . . 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 3993 . . . . . . . 8 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑗 ∈ ℤ)
1716zred 12720 . . . . . . 7 (((((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑗) = 𝑦) → 𝑗 ∈ ℝ)
1814, 17lttri4d 11400 . . . . . 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 791 . . . . . . . . . 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 792 . . . . . . . . . 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 776 . . . . . . . . . 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 13700 . . . . . . . . . . . 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 13699 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑗) ↔ (𝑖 ∈ (ℤ‘0) ∧ 𝑗 ∈ ℤ ∧ 𝑖 < 𝑗))
2723, 24, 25, 26syl3anbrc 1342 . . . . . . . . . 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 32987 . . . . . . . . 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 784 . . . . . . . . 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 769 . . . . . . . . 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 5179 . . . . . . . 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 6911 . . . . . . . . 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 784 . . . . . . . . 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 769 . . . . . . . . 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 2783 . . . . . . . 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 791 . . . . . . . . . 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 792 . . . . . . . . . 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 13700 . . . . . . . . . . . 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 13699 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑖) ↔ (𝑗 ∈ (ℤ‘0) ∧ 𝑖 ∈ ℤ ∧ 𝑗 < 𝑖))
4743, 44, 45, 46syl3anbrc 1342 . . . . . . . . . 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 32987 . . . . . . . . 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 769 . . . . . . . . 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 784 . . . . . . . . 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 5179 . . . . . . . 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 1443 . . . . . 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 6738 . . . . . . 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 776 . . . . . 6 (((((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶𝑖) = 𝑥) → 𝑦 ∈ ran 𝐶)
58 fvelrnb 6969 . . . . . . 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 3160 . . . 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 769 . . . . 5 (((( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝑥 ∈ ran 𝐶)
64 fvelrnb 6969 . . . . . 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 3160 . . 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 5631 1 (( < Po 𝐴𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  wss 3963  {csn 4631   class class class wbr 5148   Po wpo 5595   Or wor 5596  dom cdm 5689  ran crn 5690   Fn wfn 6558  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   < clt 11293  cmin 11490  cz 12611  cuz 12876  ..^cfzo 13691  chash 14366  Word cword 14549  Chaincchn 32979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-chn 32980
This theorem is referenced by: (None)
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