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Theorem chnso 18647

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 2762	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → (♯‘𝐶) = (♯‘𝐶))
2		ischn 18630	. . . . . . 7 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
3	2	bilani 508	. . . . . 6 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
4	3	simpld 498	. . . . 5 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶 ∈ Word 𝐴)
5	1, 4	wrdfd 14526	. . . 4 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶:(0..^(♯‘𝐶))⟶𝐴)
6	5	frnd 6695	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → ran 𝐶 ⊆ 𝐴)
7		simpl 486	. . 3 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Po 𝐴)
8		poss 5553	. . 3 ⊢ (ran 𝐶 ⊆ 𝐴 → ( < Po 𝐴 → < Po ran 𝐶))
9	6, 7, 8	sylc 65	. 2 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Po ran 𝐶)
10		fzossz 13679	. . . . . . . . 9 ⊢ (0..^(♯‘𝐶)) ⊆ ℤ
11		simp-4r 793	. . . . . . . . 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)))
12	10, 11	sselid 3932	. . . . . . . 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)
13	12	zred 12671	. . . . . . 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)
14		simplr 778	. . . . . . . . 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)))
15	10, 14	sselid 3932	. . . . . . . 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)
16	15	zred 12671	. . . . . . 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)
17	13, 16	lttri4d 11318	. . . . . 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))
18		simp-8l 800	. . . . . . . . . 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)
19		simp-8r 801	. . . . . . . . . 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))
20		simpllr 785	. . . . . . . . . 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)))
21		elfzouz 13663	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝐶)) → 𝑖 ∈ (ℤ_≥‘0))
22	21	ad5antlr 745	. . . . . . . . . . 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))
23	15	adantr 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) → 𝑗 ∈ ℤ)
24		simpr 488	. . . . . . . . . . 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)
25		elfzo2 13661	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑗) ↔ (𝑖 ∈ (ℤ_≥‘0) ∧ 𝑗 ∈ ℤ ∧ 𝑖 < 𝑗))
26	22, 23, 24, 25	syl3anbrc 1356	. . . . . . . . . 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))
27	18, 19, 20, 26	chnlt 18646	. . . . . . . . 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))
28		simp-4r 793	. . . . . . . . 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)
29		simplr 778	. . . . . . . . 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)
30	27, 28, 29	3brtr3d 5128	. . . . . . . 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)
31	30	ex 416	. . . . . . 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))
32		simpr 488	. . . . . . . . . 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)
33	32	fveq2d 6866	. . . . . . . . 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))
34		simp-4r 793	. . . . . . . . 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)
35		simplr 778	. . . . . . . . 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)
36	33, 34, 35	3eqtr3d 2804	. . . . . . . 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)
37	36	ex 416	. . . . . . 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))
38		simp-8l 800	. . . . . . . . . 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)
39		simp-8r 801	. . . . . . . . . 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))
40	11	adantr 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) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0..^(♯‘𝐶)))
41		elfzouz 13663	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^(♯‘𝐶)) → 𝑗 ∈ (ℤ_≥‘0))
42	41	ad3antlr 741	. . . . . . . . . . 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))
43	12	adantr 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) → 𝑖 ∈ ℤ)
44		simpr 488	. . . . . . . . . . 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)
45		elfzo2 13661	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑖) ↔ (𝑗 ∈ (ℤ_≥‘0) ∧ 𝑖 ∈ ℤ ∧ 𝑗 < 𝑖))
46	42, 43, 44, 45	syl3anbrc 1356	. . . . . . . . . 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))
47	38, 39, 40, 46	chnlt 18646	. . . . . . . . 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))
48		simplr 778	. . . . . . . . 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)
49		simp-4r 793	. . . . . . . . 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)
50	47, 48, 49	3brtr3d 5128	. . . . . . . 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)
51	50	ex 416	. . . . . . 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))
52	31, 37, 51	3orim123d 1464	. . . . . 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)))
53	17, 52	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))
54	5	ffnd 6687	. . . . . . 7 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → 𝐶 Fn (0..^(♯‘𝐶)))
55	54	ad4antr 742	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝐶 Fn (0..^(♯‘𝐶)))
56		simpllr 785	. . . . . 6 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → 𝑦 ∈ ran 𝐶)
57		fvelrnb 6922	. . . . . . 7 ⊢ (𝐶 Fn (0..^(♯‘𝐶)) → (𝑦 ∈ ran 𝐶 ↔ ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦))
58	57	biimpa 480	. . . . . 6 ⊢ ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦)
59	55, 56, 58	syl2anc 593	. . . . 5 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → ∃𝑗 ∈ (0..^(♯‘𝐶))(𝐶‘𝑗) = 𝑦)
60	53, 59	r19.29a 3169	. . . 4 ⊢ (((((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) ∧ 𝑖 ∈ (0..^(♯‘𝐶))) ∧ (𝐶‘𝑖) = 𝑥) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
61	54	ad2antrr 736	. . . . 5 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝐶 Fn (0..^(♯‘𝐶)))
62		simplr 778	. . . . 5 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → 𝑥 ∈ ran 𝐶)
63		fvelrnb 6922	. . . . . 6 ⊢ (𝐶 Fn (0..^(♯‘𝐶)) → (𝑥 ∈ ran 𝐶 ↔ ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥))
64	63	biimpa 480	. . . . 5 ⊢ ((𝐶 Fn (0..^(♯‘𝐶)) ∧ 𝑥 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥)
65	61, 62, 64	syl2anc 593	. . . 4 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → ∃𝑖 ∈ (0..^(♯‘𝐶))(𝐶‘𝑖) = 𝑥)
66	60, 65	r19.29a 3169	. . 3 ⊢ (((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ 𝑥 ∈ ran 𝐶) ∧ 𝑦 ∈ ran 𝐶) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
67	66	anasss 470	. 2 ⊢ ((( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) ∧ (𝑥 ∈ ran 𝐶 ∧ 𝑦 ∈ ran 𝐶)) → (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥))
68	9, 67	issod 5586	1 ⊢ (( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain 𝐴)) → < Or ran 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ w3o 1096 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ∖ cdif 3899 ⊆ wss 3902 {csn 4579 class class class wbr 5097 Po wpo 5549 Or wor 5550 dom cdm 5643 ran crn 5644 Fn wfn 6511 ‘cfv 6516 (class class class)co 7391 0cc0 11067 1c1 11068 < clt 11210 − cmin 11408 ℤcz 12562 ℤ_≥cuz 12833 ..^cfzo 13653 ♯chash 14337 Word cword 14520 Chain cchn 18628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-n0 12476 df-xnn0 12549 df-z 12563 df-uz 12834 df-rp 12988 df-fz 13507 df-fzo 13654 df-hash 14338 df-word 14521 df-lsw 14570 df-concat 14578 df-s1 14604 df-substr 14649 df-pfx 14679 df-chn 18629

This theorem is referenced by: (None)

W3C validator