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Theorem noprefixmo 33316
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 3323 . . . 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 777 . . . . . . . . . 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 776 . . . . . . . . . 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 4473 . . . . . . . . 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 4434 . . . . . . . . . . . 12 (𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑝)
65adantr 484 . . . . . . . . . . 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 766 . . . . . . . . . . . . . 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 3919 . . . . . . . . . . . . 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 3919 . . . . . . . . . . . . 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 33295 . . . . . . . . . . . . . 14 <s Or No
11 soasym 5472 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑢 No 𝑝 No )) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
1210, 11mpan 689 . . . . . . . . . . . . 13 ((𝑢 No 𝑝 No ) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
138, 9, 12syl2anc 587 . . . . . . . . . . . 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 411 . . . . . . . . . . 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 5051 . . . . . . . . . 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 4437 . . . . . . . . . . . 12 𝑢 <s 𝑝 → if(𝑢 <s 𝑝, 𝑝, 𝑢) = 𝑢)
1716adantr 484 . . . . . . . . . . 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 5464 . . . . . . . . . . . . . 14 (( <s Or No 𝑢 No ) → ¬ 𝑢 <s 𝑢)
1910, 18mpan 689 . . . . . . . . . . . . 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 485 . . . . . . . . . . 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 5051 . . . . . . . . . 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 811 . . . . . . . . 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 5464 . . . . . . . . . . . . . 14 (( <s Or No 𝑝 No ) → ¬ 𝑝 <s 𝑝)
2510, 24mpan 689 . . . . . . . . . . . . 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 485 . . . . . . . . . . 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 5051 . . . . . . . . . 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 486 . . . . . . . . . . 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 5051 . . . . . . . . . 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 811 . . . . . . . . 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 1191 . . . . . . . . . . . 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 1216 . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 1192 . . . . . . . . . . . 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 5036 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑢 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
3736notbid 321 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑢 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑢))
38 reseq1 5816 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 ↾ suc 𝐺) = (if(𝑢 <s 𝑝, 𝑝, 𝑢) ↾ suc 𝐺))
3938eqeq2d 2812 . . . . . . . . . . . . . 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 348 . . . . . . . . . . . . 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 3569 . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 5036 . . . . . . . . . . . . . . 15 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (𝑣 <s 𝑝 ↔ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4746notbid 321 . . . . . . . . . . . . . 14 (𝑣 = if(𝑢 <s 𝑝, 𝑝, 𝑢) → (¬ 𝑣 <s 𝑝 ↔ ¬ if(𝑢 <s 𝑝, 𝑝, 𝑢) <s 𝑝))
4838eqeq2d 2812 . . . . . . . . . . . . . 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 348 . . . . . . . . . . . . 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 3569 . . . . . . . . . . . 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 2839 . . . . . . . . . 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 416 . . . . . . . . 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 1461 . . . . . . . 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 6651 . . . . . . 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 1215 . . . . . . . . . 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 6241 . . . . . . . . . 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 6669 . . . . . . . 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 1217 . . . . . . . 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 2836 . . . . . . 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 6669 . . . . . . . 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 1220 . . . . . . . 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 2836 . . . . . . 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 2844 . . . . . 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 416 . . . . 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 3256 . . . 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 246 . . 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 1929 . 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 2813 . . . . . 6 (𝑥 = 𝑦 → ((𝑢𝐺) = 𝑥 ↔ (𝑢𝐺) = 𝑦))
71703anbi3d 1439 . . . . 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 3259 . . . 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 5740 . . . . . . 7 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
7473eleq2d 2878 . . . . . 6 (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢𝐺 ∈ dom 𝑝))
75 breq2 5037 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
7675notbid 321 . . . . . . . 8 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
77 reseq1 5816 . . . . . . . . 9 (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
7877eqeq1d 2803 . . . . . . . 8 (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
7976, 78imbi12d 348 . . . . . . 7 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
8079ralbidv 3165 . . . . . 6 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
81 fveq1 6648 . . . . . . 7 (𝑢 = 𝑝 → (𝑢𝐺) = (𝑝𝐺))
8281eqeq1d 2803 . . . . . 6 (𝑢 = 𝑝 → ((𝑢𝐺) = 𝑦 ↔ (𝑝𝐺) = 𝑦))
8374, 80, 823anbi123d 1433 . . . . 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)))
8483cbvrexvw 3400 . . . 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 290 . . 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 2628 . 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 237 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 399  w3a 1084  wal 1536   = wceq 1538  wcel 2112  ∃*wmo 2599  wral 3109  wrex 3110  wss 3884  ifcif 4428   class class class wbr 5033   Or wor 5441  dom cdm 5523  cres 5525  suc csuc 6165  cfv 6328   No csur 33261   <s cslt 33262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-1o 8089  df-2o 8090  df-no 33264  df-slt 33265
This theorem is referenced by:  nosupno  33317  nosupfv  33320
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