Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  noprefixmo Structured version   Visualization version   GIF version

Theorem noprefixmo 31973
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 3136 . . . 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 818 . . . . . . . . . 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 817 . . . . . . . . . 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 4164 . . . . . . . . 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 4125 . . . . . . . . . . . 12 (𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑝)
65adantr 480 . . . . . . . . . . 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 805 . . . . . . . . . . . . . 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 3637 . . . . . . . . . . . . 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 3637 . . . . . . . . . . . . 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 31952 . . . . . . . . . . . . . 14 <s Or No
11 soasym 31783 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑢 No 𝑝 No )) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
1210, 11mpan 706 . . . . . . . . . . . . 13 ((𝑢 No 𝑝 No ) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
138, 9, 12syl2anc 694 . . . . . . . . . . . 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 445 . . . . . . . . . . 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 4703 . . . . . . . . . 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 4128 . . . . . . . . . . . 12 𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑢)
1716adantr 480 . . . . . . . . . . 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 5085 . . . . . . . . . . . . . 14 (( <s Or No 𝑢 No ) → ¬ 𝑢 <s 𝑢)
1910, 18mpan 706 . . . . . . . . . . . . 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 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)))) → ¬ 𝑢 <s 𝑢)
2217, 21eqnbrtrd 4703 . . . . . . . . . 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 848 . . . . . . . . 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 5085 . . . . . . . . . . . . . 14 (( <s Or No 𝑝 No ) → ¬ 𝑝 <s 𝑝)
2510, 24mpan 706 . . . . . . . . . . . . 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 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)))) → ¬ 𝑝 <s 𝑝)
286, 27eqnbrtrd 4703 . . . . . . . . . 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 472 . . . . . . . . . . 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 4703 . . . . . . . . . 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 848 . . . . . . . . 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 1087 . . . . . . . . . . . 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 1127 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 1088 . . . . . . . . . . . 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 4688 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑢 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
3736notbid 307 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑢 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
38 reseq1 5422 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))
3938eqeq2d 2661 . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . 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 3336 . . . . . . . . . . . 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 1130 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 1089 . . . . . . . . . . . 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 4688 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑝 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4746notbid 307 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑝 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4838eqeq2d 2661 . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . 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 3336 . . . . . . . . . . . 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 2688 . . . . . . . . . 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 449 . . . . . . . . 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 1467 . . . . . . . 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 6231 . . . . . . 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 1126 . . . . . . . . . 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 5841 . . . . . . . . . 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 6246 . . . . . . . 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 1128 . . . . . . . 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 2685 . . . . . . 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 6246 . . . . . . . 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 1131 . . . . . . . 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 2685 . . . . . . 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 2693 . . . . . 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 449 . . . . 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 3067 . . . 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 1896 . 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 2662 . . . . . 6 (𝑥 = 𝑦 → ((𝑢𝐺) = 𝑥 ↔ (𝑢𝐺) = 𝑦))
71703anbi3d 1445 . . . . 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 3081 . . . 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 5356 . . . . . . 7 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
7473eleq2d 2716 . . . . . 6 (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢𝐺 ∈ dom 𝑝))
75 breq2 4689 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
7675notbid 307 . . . . . . . 8 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
77 reseq1 5422 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
7877eqeq1d 2653 . . . . . . . 8 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
7976, 78imbi12d 333 . . . . . . 7 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
8079ralbidv 3015 . . . . . 6 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
81 fveq1 6228 . . . . . . 7 (𝑢 = 𝑝 → (𝑢𝐺) = (𝑝𝐺))
8281eqeq1d 2653 . . . . . 6 (𝑢 = 𝑝 → ((𝑢𝐺) = 𝑦 ↔ (𝑝𝐺) = 𝑦))
8374, 80, 823anbi123d 1439 . . . . 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 3202 . . . 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 2546 . 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 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wral 2941  wrex 2942  wss 3607  ifcif 4119   class class class wbr 4685   Or wor 5063  dom cdm 5143  cres 5145  suc csuc 5763  cfv 5926   No csur 31918   <s cslt 31919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922
This theorem is referenced by:  nosupno  31974  nosupfv  31977
  Copyright terms: Public domain W3C validator