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Theorem noprefixmo 31822
Description: In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.)
Assertion
Ref Expression
noprefixmo (𝐴 No → ∃*𝑥𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑢,𝐺,𝑣,𝑥

Proof of Theorem noprefixmo
Dummy variables 𝑦 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 3102 . . . 4 (∃𝑢𝐴𝑝𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) ↔ (∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))
2 simplrr 800 . . . . . . . . . 10 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝑝𝐴)
3 simplrl 799 . . . . . . . . . 10 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝑢𝐴)
42, 3ifcld 4122 . . . . . . . . 9 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴)
5 iftrue 4083 . . . . . . . . . . . 12 (𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑝)
65adantr 481 . . . . . . . . . . 11 ((𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑝)
7 simpll 789 . . . . . . . . . . . . . 14 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝐴 No )
87, 3sseldd 3596 . . . . . . . . . . . . 13 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝑢 No )
97, 2sseldd 3596 . . . . . . . . . . . . 13 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝑝 No )
10 sltso 31801 . . . . . . . . . . . . . 14 <s Or No
11 soasym 31633 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑢 No 𝑝 No )) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
1210, 11mpan 705 . . . . . . . . . . . . 13 ((𝑢 No 𝑝 No ) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
138, 9, 12syl2anc 692 . . . . . . . . . . . 12 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
1413impcom 446 . . . . . . . . . . 11 ((𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ 𝑝 <s 𝑢)
156, 14eqnbrtrd 4662 . . . . . . . . . 10 ((𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢)
16 iffalse 4086 . . . . . . . . . . . 12 𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑢)
1716adantr 481 . . . . . . . . . . 11 ((¬ 𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑢)
18 sonr 5046 . . . . . . . . . . . . . 14 (( <s Or No 𝑢 No ) → ¬ 𝑢 <s 𝑢)
1910, 18mpan 705 . . . . . . . . . . . . 13 (𝑢 No → ¬ 𝑢 <s 𝑢)
208, 19syl 17 . . . . . . . . . . . 12 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ¬ 𝑢 <s 𝑢)
2120adantl 482 . . . . . . . . . . 11 ((¬ 𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ 𝑢 <s 𝑢)
2217, 21eqnbrtrd 4662 . . . . . . . . . 10 ((¬ 𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢)
2315, 22pm2.61ian 830 . . . . . . . . 9 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢)
24 sonr 5046 . . . . . . . . . . . . . 14 (( <s Or No 𝑝 No ) → ¬ 𝑝 <s 𝑝)
2510, 24mpan 705 . . . . . . . . . . . . 13 (𝑝 No → ¬ 𝑝 <s 𝑝)
269, 25syl 17 . . . . . . . . . . . 12 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ¬ 𝑝 <s 𝑝)
2726adantl 482 . . . . . . . . . . 11 ((𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ 𝑝 <s 𝑝)
286, 27eqnbrtrd 4662 . . . . . . . . . 10 ((𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)
29 simpl 473 . . . . . . . . . . 11 ((¬ 𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ 𝑢 <s 𝑝)
3017, 29eqnbrtrd 4662 . . . . . . . . . 10 ((¬ 𝑢 <s 𝑝 ∧ ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)
3128, 30pm2.61ian 830 . . . . . . . . 9 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)
32 simpr1 1065 . . . . . . . . . . . 12 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴)
33 simprl2 1105 . . . . . . . . . . . . 13 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
3433adantr 481 . . . . . . . . . . . 12 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
35 simpr2 1066 . . . . . . . . . . . 12 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢)
36 breq1 4647 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑢 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
3736notbid 308 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑢 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
38 reseq1 5379 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))
3938eqeq2d 2630 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑢 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺)))
4037, 39imbi12d 334 . . . . . . . . . . . . 13 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 → (𝑢 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))))
4140rspcv 3300 . . . . . . . . . . . 12 (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → (¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 → (𝑢 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))))
4232, 34, 35, 41syl3c 66 . . . . . . . . . . 11 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → (𝑢 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))
43 simprr2 1108 . . . . . . . . . . . . 13 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
4443adantr 481 . . . . . . . . . . . 12 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
45 simpr3 1067 . . . . . . . . . . . 12 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)
46 breq1 4647 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑝 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4746notbid 308 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑝 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4838eqeq2d 2630 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → ((𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺)))
4947, 48imbi12d 334 . . . . . . . . . . . . 13 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝 → (𝑝 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))))
5049rspcv 3300 . . . . . . . . . . . 12 (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 → (∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) → (¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝 → (𝑝 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))))
5132, 44, 45, 50syl3c 66 . . . . . . . . . . 11 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → (𝑝 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))
5242, 51eqtr4d 2657 . . . . . . . . . 10 ((((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) ∧ (if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝)) → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
5352ex 450 . . . . . . . . 9 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((if(𝑢 <s 𝑝, 𝑝, 𝑢) ∈ 𝐴 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢 ∧ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝) → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
544, 23, 31, 53mp3and 1425 . . . . . . . 8 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
5554fveq1d 6180 . . . . . . 7 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = ((𝑝 ↾ suc 𝐺)‘𝐺))
56 simprl1 1104 . . . . . . . . . 10 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝐺 ∈ dom 𝑢)
57 sucidg 5791 . . . . . . . . . 10 (𝐺 ∈ dom 𝑢𝐺 ∈ suc 𝐺)
5856, 57syl 17 . . . . . . . . 9 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝐺 ∈ suc 𝐺)
5958fvresd 6195 . . . . . . . 8 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = (𝑢𝐺))
60 simprl3 1106 . . . . . . . 8 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → (𝑢𝐺) = 𝑥)
6159, 60eqtrd 2654 . . . . . . 7 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = 𝑥)
6258fvresd 6195 . . . . . . . 8 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((𝑝 ↾ suc 𝐺)‘𝐺) = (𝑝𝐺))
63 simprr3 1109 . . . . . . . 8 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → (𝑝𝐺) = 𝑦)
6462, 63eqtrd 2654 . . . . . . 7 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → ((𝑝 ↾ suc 𝐺)‘𝐺) = 𝑦)
6555, 61, 643eqtr3d 2662 . . . . . 6 (((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))) → 𝑥 = 𝑦)
6665ex 450 . . . . 5 ((𝐴 No ∧ (𝑢𝐴𝑝𝐴)) → (((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) → 𝑥 = 𝑦))
6766rexlimdvva 3034 . . . 4 (𝐴 No → (∃𝑢𝐴𝑝𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) → 𝑥 = 𝑦))
681, 67syl5bir 233 . . 3 (𝐴 No → ((∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) → 𝑥 = 𝑦))
6968alrimivv 1854 . 2 (𝐴 No → ∀𝑥𝑦((∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) → 𝑥 = 𝑦))
70 eqeq2 2631 . . . . . 6 (𝑥 = 𝑦 → ((𝑢𝐺) = 𝑥 ↔ (𝑢𝐺) = 𝑦))
71703anbi3d 1403 . . . . 5 (𝑥 = 𝑦 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ↔ (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑦)))
7271rexbidv 3048 . . . 4 (𝑥 = 𝑦 → (∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ↔ ∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑦)))
73 dmeq 5313 . . . . . . 7 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
7473eleq2d 2685 . . . . . 6 (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢𝐺 ∈ dom 𝑝))
75 breq2 4648 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
7675notbid 308 . . . . . . . 8 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
77 reseq1 5379 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
7877eqeq1d 2622 . . . . . . . 8 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
7976, 78imbi12d 334 . . . . . . 7 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
8079ralbidv 2983 . . . . . 6 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
81 fveq1 6177 . . . . . . 7 (𝑢 = 𝑝 → (𝑢𝐺) = (𝑝𝐺))
8281eqeq1d 2622 . . . . . 6 (𝑢 = 𝑝 → ((𝑢𝐺) = 𝑦 ↔ (𝑝𝐺) = 𝑦))
8374, 80, 823anbi123d 1397 . . . . 5 (𝑢 = 𝑝 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑦) ↔ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))
8483cbvrexv 3167 . . . 4 (∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑦) ↔ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦))
8572, 84syl6bb 276 . . 3 (𝑥 = 𝑦 → (∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ↔ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)))
8685mo4 2515 . 2 (∃*𝑥𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ↔ ∀𝑥𝑦((∃𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥) ∧ ∃𝑝𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝𝐺) = 𝑦)) → 𝑥 = 𝑦))
8769, 86sylibr 224 1 (𝐴 No → ∃*𝑥𝑢𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢𝐺) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  ∃*wmo 2469  wral 2909  wrex 2910  wss 3567  ifcif 4077   class class class wbr 4644   Or wor 5024  dom cdm 5104  cres 5106  suc csuc 5713  cfv 5876   No csur 31767   <s cslt 31768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-1o 7545  df-2o 7546  df-no 31770  df-slt 31771
This theorem is referenced by:  nosupno  31823  nosupfv  31826
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