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Theorem noinfbday 27671
Description: Birthday bounding law for surreal infimum. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
noinfbday.1 𝑇 = if(βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯, ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐡 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐡 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
Assertion
Ref Expression
noinfbday (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ ( bday β€˜π‘‡) βŠ† 𝑂)
Distinct variable groups:   𝐡,𝑔,𝑒,𝑣,π‘₯,𝑦   𝑔,𝑉   π‘₯,𝑣,𝑦
Allowed substitution hints:   𝑇(π‘₯,𝑦,𝑣,𝑒,𝑔)   𝑂(π‘₯,𝑦,𝑣,𝑒,𝑔)   𝑉(π‘₯,𝑦,𝑣,𝑒)
Proof of Theorem noinfbday
Dummy variables 𝑝 π‘ž 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noinfbday.1 . . . . 5 𝑇 = if(βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯, ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐡 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐡 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯))))
21noinfno 27669 . . . 4 ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) β†’ 𝑇 ∈ No )
3 bdayval 27599 . . . 4 (𝑇 ∈ No β†’ ( bday β€˜π‘‡) = dom 𝑇)
42, 3syl 17 . . 3 ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) β†’ ( bday β€˜π‘‡) = dom 𝑇)
54adantr 479 . 2 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ ( bday β€˜π‘‡) = dom 𝑇)
6 iftrue 4530 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ if(βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯, ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}), (𝑔 ∈ {𝑦 ∣ βˆƒπ‘’ ∈ 𝐡 (𝑦 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑦) = (𝑣 β†Ύ suc 𝑦)))} ↦ (β„©π‘₯βˆƒπ‘’ ∈ 𝐡 (𝑔 ∈ dom 𝑒 ∧ βˆ€π‘£ ∈ 𝐡 (Β¬ 𝑒 <s 𝑣 β†’ (𝑒 β†Ύ suc 𝑔) = (𝑣 β†Ύ suc 𝑔)) ∧ (π‘’β€˜π‘”) = π‘₯)))) = ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}))
71, 6eqtrid 2777 . . . . . . 7 (βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ 𝑇 = ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}))
87dmeqd 5902 . . . . . 6 (βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ dom 𝑇 = dom ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}))
9 1oex 8495 . . . . . . . . 9 1o ∈ V
109dmsnop 6215 . . . . . . . 8 dom {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩} = {dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)}
1110uneq2i 4153 . . . . . . 7 (dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ dom {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}) = (dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)})
12 dmun 5907 . . . . . . 7 dom ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}) = (dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ dom {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩})
13 df-suc 6370 . . . . . . 7 suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) = (dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)})
1411, 12, 133eqtr4i 2763 . . . . . 6 dom ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βˆͺ {⟨dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯), 1o⟩}) = suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)
158, 14eqtrdi 2781 . . . . 5 (βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ dom 𝑇 = suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯))
1615adantr 479 . . . 4 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ dom 𝑇 = suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯))
17 simprrl 779 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ 𝑂 ∈ On)
18 eloni 6374 . . . . . 6 (𝑂 ∈ On β†’ Ord 𝑂)
1917, 18syl 17 . . . . 5 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ Ord 𝑂)
20 simprll 777 . . . . . . . 8 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ 𝐡 βŠ† No )
21 simpl 481 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)
22 nominmo 27650 . . . . . . . . . . 11 (𝐡 βŠ† No β†’ βˆƒ*π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)
2320, 22syl 17 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ βˆƒ*π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)
24 reu5 3366 . . . . . . . . . 10 (βˆƒ!π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ↔ (βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ βˆƒ*π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯))
2521, 23, 24sylanbrc 581 . . . . . . . . 9 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ βˆƒ!π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)
26 riotacl 7390 . . . . . . . . 9 (βˆƒ!π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ 𝐡)
2725, 26syl 17 . . . . . . . 8 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ 𝐡)
2820, 27sseldd 3973 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ No )
29 bdayval 27599 . . . . . . 7 ((β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ No β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)) = dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯))
3028, 29syl 17 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)) = dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯))
31 simprrr 780 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ ( bday β€œ 𝐡) βŠ† 𝑂)
32 bdayfo 27628 . . . . . . . . 9 bday : No –ontoβ†’On
33 fofn 6808 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3432, 33ax-mp 5 . . . . . . . 8 bday Fn No
35 fnfvima 7241 . . . . . . . 8 (( bday Fn No ∧ 𝐡 βŠ† No ∧ (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ 𝐡) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)) ∈ ( bday β€œ 𝐡))
3634, 20, 27, 35mp3an2i 1462 . . . . . . 7 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)) ∈ ( bday β€œ 𝐡))
3731, 36sseldd 3973 . . . . . 6 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ ( bday β€˜(β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯)) ∈ 𝑂)
3830, 37eqeltrrd 2826 . . . . 5 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ 𝑂)
39 ordsucss 7819 . . . . 5 (Ord 𝑂 β†’ (dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) ∈ 𝑂 β†’ suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βŠ† 𝑂))
4019, 38, 39sylc 65 . . . 4 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ suc dom (β„©π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯) βŠ† 𝑂)
4116, 40eqsstrd 4011 . . 3 ((βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ dom 𝑇 βŠ† 𝑂)
421noinfdm 27670 . . . . 5 (Β¬ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ β†’ dom 𝑇 = {𝑧 ∣ βˆƒπ‘ ∈ 𝐡 (𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧)))})
4342adantr 479 . . . 4 ((Β¬ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ dom 𝑇 = {𝑧 ∣ βˆƒπ‘ ∈ 𝐡 (𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧)))})
44 simplrl 775 . . . . . . . . . . 11 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ 𝑂 ∈ On)
4544, 18syl 17 . . . . . . . . . 10 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ Ord 𝑂)
46 ssel2 3967 . . . . . . . . . . . . 13 ((𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ 𝑝 ∈ No )
4746ad4ant14 750 . . . . . . . . . . . 12 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ 𝑝 ∈ No )
48 bdayval 27599 . . . . . . . . . . . 12 (𝑝 ∈ No β†’ ( bday β€˜π‘) = dom 𝑝)
4947, 48syl 17 . . . . . . . . . . 11 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) = dom 𝑝)
50 simplrr 776 . . . . . . . . . . . 12 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€œ 𝐡) βŠ† 𝑂)
51 fnfvima 7241 . . . . . . . . . . . . . 14 (( bday Fn No ∧ 𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ ( bday β€œ 𝐡))
5234, 51mp3an1 1444 . . . . . . . . . . . . 13 ((𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ ( bday β€œ 𝐡))
5352ad4ant14 750 . . . . . . . . . . . 12 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ ( bday β€œ 𝐡))
5450, 53sseldd 3973 . . . . . . . . . . 11 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ 𝑂)
5549, 54eqeltrrd 2826 . . . . . . . . . 10 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ dom 𝑝 ∈ 𝑂)
56 ordelss 6380 . . . . . . . . . 10 ((Ord 𝑂 ∧ dom 𝑝 ∈ 𝑂) β†’ dom 𝑝 βŠ† 𝑂)
5745, 55, 56syl2anc 582 . . . . . . . . 9 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ dom 𝑝 βŠ† 𝑂)
5857sseld 3971 . . . . . . . 8 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ (𝑧 ∈ dom 𝑝 β†’ 𝑧 ∈ 𝑂))
5958adantrd 490 . . . . . . 7 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ((𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧))) β†’ 𝑧 ∈ 𝑂))
6059rexlimdva 3145 . . . . . 6 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ (βˆƒπ‘ ∈ 𝐡 (𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧))) β†’ 𝑧 ∈ 𝑂))
6160abssdv 4057 . . . . 5 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ {𝑧 ∣ βˆƒπ‘ ∈ 𝐡 (𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧)))} βŠ† 𝑂)
6261adantl 480 . . . 4 ((Β¬ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ {𝑧 ∣ βˆƒπ‘ ∈ 𝐡 (𝑧 ∈ dom 𝑝 ∧ βˆ€π‘ž ∈ 𝐡 (Β¬ 𝑝 <s π‘ž β†’ (𝑝 β†Ύ suc 𝑧) = (π‘ž β†Ύ suc 𝑧)))} βŠ† 𝑂)
6343, 62eqsstrd 4011 . . 3 ((Β¬ βˆƒπ‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 <s π‘₯ ∧ ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂))) β†’ dom 𝑇 βŠ† 𝑂)
6441, 63pm2.61ian 810 . 2 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ dom 𝑇 βŠ† 𝑂)
655, 64eqsstrd 4011 1 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ ( bday β€˜π‘‡) βŠ† 𝑂)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆ€wral 3051  βˆƒwrex 3060  βˆƒ!wreu 3362  βˆƒ*wrmo 3363   βˆͺ cun 3937   βŠ† wss 3939  ifcif 4524  {csn 4624  βŸ¨cop 4630   class class class wbr 5143   ↦ cmpt 5226  dom cdm 5672   β†Ύ cres 5674   β€œ cima 5675  Ord word 6363  Oncon0 6364  suc csuc 6366  β„©cio 6493   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  β„©crio 7371  1oc1o 8478   No csur 27591   <s cslt 27592   bday cbday 27593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-1o 8485  df-2o 8486  df-no 27594  df-slt 27595  df-bday 27596
This theorem is referenced by:  noetalem1  27692
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