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Theorem chnso 18534

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

**Assertion**
Ref	Expression
chnso	⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < 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 2734	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → (♯‘𝐶) = (♯‘𝐶))
2		ischn 18517	. . . . . . . 8 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	biimpi 216	. . . . . . 7 ⊢ (𝐶 ∈ ( < Chain 𝐴) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
4	3	adantl 481	. . . . . 6 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
5	4	simpld 494	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶 ∈ Word 𝐴)
6	1, 5	wrdfd 14430	. . . 4 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶:(0..^(♯‘𝐶))⟶𝐴)
7	6	frnd 6666	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → ran 𝐶 ⊆ 𝐴)
8		simpl 482	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Po 𝐴)
9		poss 5531	. . 3 ⊢ (ran 𝐶 ⊆ 𝐴 → ( < Po 𝐴 → < Po ran 𝐶))
10	7, 8, 9	sylc 65	. 2 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Po ran 𝐶)
11		fzossz 13583	. . . . . . . . 9 ⊢ (0..^(♯‘𝐶)) ⊆ ℤ
12		simp-4r 783	. . . . . . . . 9 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 3928	. . . . . . . 8 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑖 ∈ ℤ)
14	13	zred 12585	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 3928	. . . . . . . 8 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑗 ∈ ℤ)
17	16	zred 12585	. . . . . . 7 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) → 𝑗 ∈ ℝ)
18	14, 17	lttri4d 11263	. . . . . 6 ⊢ (((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝐶 ∈ ( < Chain 𝐴))
21		simpllr 775	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 13567	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝐶)) → 𝑖 ∈ (ℤ_≥‘0))
23	22	ad5antlr 735	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
26		elfzo2 13566	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑗) ↔ (𝑖 ∈ (ℤ_≥‘0) ∧ 𝑗 ∈ ℤ ∧ 𝑖 < 𝑗))
27	23, 24, 25, 26	syl3anbrc 1344	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 18533	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 5126	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 6834	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 2776	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝐶 ∈ ( < Chain 𝐴))
41	12	adantr 480	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 13567	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^(♯‘𝐶)) → 𝑗 ∈ (ℤ_≥‘0))
43	42	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) ∧ 𝑗 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑗) = 𝑦) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
46		elfzo2 13566	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑖) ↔ (𝑗 ∈ (ℤ_≥‘0) ∧ 𝑖 ∈ ℤ ∧ 𝑗 < 𝑖))
47	43, 44, 45, 46	syl3anbrc 1344	. . . . . . . . . 10 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 18533	. . . . . . . . 9 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 5126	. . . . . . . 8 ⊢ ((((((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 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 6659	. . . . . . 7 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶 Fn (0..^(♯‘𝐶)))
56	55	ad4antr 732	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝐶 Fn (0..^(♯‘𝐶)))
57		simpllr 775	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝑦 ∈ ran 𝐶)
58		fvelrnb 6890	. . . . . . 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦)
61	54, 60	r19.29a 3141	. . . 4 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝐶 Fn (0..^(♯‘𝐶)))
63		simplr 768	. . . . 5 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝑥 ∈ ran 𝐶)
64		fvelrnb 6890	. . . . . 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 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥)
67	61, 66	r19.29a 3141	. . 3 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
68	67	anasss 466	. 2 ⊢ ((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ (𝑥 ∈ ran 𝐶 ∧ 𝑦 ∈ ran 𝐶)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
69	10, 68	issod 5564	1 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Or ran 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 ∖ cdif 3895 ⊆ wss 3898 {csn 4577 class class class wbr 5095 Po wpo 5527 Or wor 5528 dom cdm 5621 ran crn 5622 Fn wfn 6483 ‘cfv 6488 (class class class)co 7354 0cc0 11015 1c1 11016 < clt 11155 − cmin 11353 ℤcz 12477 ℤ_≥cuz 12740 ..^cfzo 13558 ♯chash 14241 Word cword 14424 Chain cchn 18515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7805 df-1st 7929 df-2nd 7930 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-n0 12391 df-xnn0 12464 df-z 12478 df-uz 12741 df-rp 12895 df-fz 13412 df-fzo 13559 df-hash 14242 df-word 14425 df-lsw 14474 df-concat 14482 df-s1 14508 df-substr 14553 df-pfx 14583 df-chn 18516

This theorem is referenced by: (None)

W3C validator