Theorem chnso 32999

Description: A chain induces a total order. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Assertion

Ref	Expression
chnso	⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)

Proof of Theorem chnso

Dummy variables 𝑥 𝑦 𝑖 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → (♯‘𝐶) = (♯‘𝐶))
2		ischn 32991	. . . . . . . 8 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	biimpi 216	. . . . . . 7 ⊢ (𝐶 ∈ ( < Chain𝐴) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
4	3	adantl 481	. . . . . 6 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
5	4	simpld 494	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → 𝐶 ∈ Word 𝐴)
6	1, 5	wrdfd 14542	. . . 4 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → 𝐶:(0..^(♯‘𝐶))⟶𝐴)
7	6	frnd 6719	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → ran 𝐶 ⊆ 𝐴)
8		simpl 482	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → < Po 𝐴)
9		poss 5568	. . 3 ⊢ (ran 𝐶 ⊆ 𝐴 → ( < Po 𝐴 → < Po ran 𝐶))
10	7, 8, 9	sylc 65	. 2 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → < Po ran 𝐶)
11		fzossz 13701	. . . . . . . . 9 ⊢ (0..^(♯‘𝐶)) ⊆ ℤ
12		simp-4r 783	. . . . . . . . 9 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑖 ∈ (0..^(♯‘𝐶)))
13	11, 12	sselid 3961	. . . . . . . 8 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑖 ∈ ℤ)
14	13	zred 12702	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑖 ∈ ℝ)
15		simplr 768	. . . . . . . . 9 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑗 ∈ (0..^(♯‘𝐶)))
16	11, 15	sselid 3961	. . . . . . . 8 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑗 ∈ ℤ)
17	16	zred 12702	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑗 ∈ ℝ)
18	14, 17	lttri4d 11381	. . . . . 6 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → (𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖))
19		simp-8l 790	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → < Po 𝐴)
20		simp-8r 791	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝐶 ∈ ( < Chain𝐴))
21		simpllr 775	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0..^(♯‘𝐶)))
22		elfzouz 13685	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝐶)) → 𝑖 ∈ (ℤ≥‘0))
23	22	ad5antlr 735	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (ℤ≥‘0))
24	16	adantr 480	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
25		simpr 484	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
26		elfzo2 13684	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑗) ↔ (𝑖 ∈ (ℤ≥‘0) ∧ 𝑗 ∈ ℤ ∧ 𝑖 < 𝑗))
27	23, 24, 25, 26	syl3anbrc 1344	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0..^𝑗))
28	19, 20, 21, 27	chnlt 32998	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶‘𝑖) < (𝐶‘𝑗))
29		simp-4r 783	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶‘𝑖) = 𝑥)
30		simplr 768	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → (𝐶‘𝑗) = 𝑦)
31	28, 29, 30	3brtr3d 5155	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑥 < 𝑦)
32	31	ex 412	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → (𝑖 < 𝑗 → 𝑥 < 𝑦))
33		simpr 484	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
34	33	fveq2d 6885	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶‘𝑖) = (𝐶‘𝑗))
35		simp-4r 783	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶‘𝑖) = 𝑥)
36		simplr 768	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → (𝐶‘𝑗) = 𝑦)
37	34, 35, 36	3eqtr3d 2779	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 = 𝑗) → 𝑥 = 𝑦)
38	37	ex 412	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → (𝑖 = 𝑗 → 𝑥 = 𝑦))
39		simp-8l 790	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → < Po 𝐴)
40		simp-8r 791	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝐶 ∈ ( < Chain𝐴))
41	12	adantr 480	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0..^(♯‘𝐶)))
42		elfzouz 13685	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^(♯‘𝐶)) → 𝑗 ∈ (ℤ≥‘0))
43	42	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (ℤ≥‘0))
44	13	adantr 480	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑖 ∈ ℤ)
45		simpr 484	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
46		elfzo2 13684	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑖) ↔ (𝑗 ∈ (ℤ≥‘0) ∧ 𝑖 ∈ ℤ ∧ 𝑗 < 𝑖))
47	43, 44, 45, 46	syl3anbrc 1344	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0..^𝑖))
48	39, 40, 41, 47	chnlt 32998	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶‘𝑗) < (𝐶‘𝑖))
49		simplr 768	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶‘𝑗) = 𝑦)
50		simp-4r 783	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → (𝐶‘𝑖) = 𝑥)
51	48, 49, 50	3brtr3d 5155	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑦 < 𝑥)
52	51	ex 412	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → (𝑗 < 𝑖 → 𝑦 < 𝑥))
53	32, 38, 52	3orim123d 1446	. . . . . 6 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → ((𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
54	18, 53	mpd 15	. . . . 5 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
55	6	ffnd 6712	. . . . . . 7 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → 𝐶 Fn (0..^(♯‘𝐶)))
56	55	ad4antr 732	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝐶 Fn (0..^(♯‘𝐶)))
57		simpllr 775	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝑦 ∈ ran 𝐶)
58		fvelrnb 6944	. . . . . . 7 ⊢ (𝐶 Fn (0..^(♯‘𝐶)) → (𝑦 ∈ ran 𝐶 ↔ ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦))
59	58	biimpa 476	. . . . . 6 ⊢ ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦)
60	56, 57, 59	syl2anc 584	. . . . 5 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦)
61	54, 60	r19.29a 3149	. . . 4 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
62	55	ad2antrr 726	. . . . 5 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝐶 Fn (0..^(♯‘𝐶)))
63		simplr 768	. . . . 5 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝑥 ∈ ran 𝐶)
64		fvelrnb 6944	. . . . . 6 ⊢ (𝐶 Fn (0..^(♯‘𝐶)) → (𝑥 ∈ ran 𝐶 ↔ ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥))
65	64	biimpa 476	. . . . 5 ⊢ ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑥 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥)
66	62, 63, 65	syl2anc 584	. . . 4 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥)
67	61, 66	r19.29a 3149	. . 3 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
68	67	anasss 466	. 2 ⊢ ((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) ∧ (𝑥 ∈ ran 𝐶 ∧ 𝑦 ∈ ran 𝐶)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
69	10, 68	issod 5601	1 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ∖ cdif 3928   ⊆ wss 3931  {csn 4606   class class class wbr 5124   Po wpo 5564   Or wor 5565  dom cdm 5659  ran crn 5660   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  0cc0 11134  1c1 11135   < clt 11274   − cmin 11471  ℤcz 12593  ℤ_≥cuz 12857  ..^cfzo 13676  ♯chash 14353  Word cword 14536  Chaincchn 32989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-xnn0 12580  df-z 12594  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-chn 32990

This theorem is referenced by: (None)