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Theorem noinfbday 27084
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 27082 . . . 4 ((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) β†’ 𝑇 ∈ No )
3 bdayval 27012 . . . 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 482 . 2 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ ( bday β€˜π‘‡) = dom 𝑇)
6 iftrue 4493 . . . . . . . 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 2785 . . . . . . 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 5862 . . . . . 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 8423 . . . . . . . . 9 1o ∈ V
109dmsnop 6169 . . . . . . . 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 4121 . . . . . . 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 5867 . . . . . . 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 6324 . . . . . . 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 2771 . . . . . 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 2789 . . . . 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 482 . . . 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 780 . . . . . 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 6328 . . . . . 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 778 . . . . . . . 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 484 . . . . . . . . . 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 27063 . . . . . . . . . . 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 3354 . . . . . . . . . 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 584 . . . . . . . . 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 7332 . . . . . . . . 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 3946 . . . . . . 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 27012 . . . . . . 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 781 . . . . . . 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 27041 . . . . . . . . 9 bday : No –ontoβ†’On
33 fofn 6759 . . . . . . . . 9 ( bday : No –ontoβ†’On β†’ bday Fn No )
3432, 33ax-mp 5 . . . . . . . 8 bday Fn No
35 fnfvima 7184 . . . . . . . 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 1467 . . . . . . 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 3946 . . . . . 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 2835 . . . . 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 7754 . . . . 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 3983 . . 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 27083 . . . . 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 482 . . . 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 776 . . . . . . . . . . 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 3940 . . . . . . . . . . . . 13 ((𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ 𝑝 ∈ No )
4746ad4ant14 751 . . . . . . . . . . . 12 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ 𝑝 ∈ No )
48 bdayval 27012 . . . . . . . . . . . 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 777 . . . . . . . . . . . 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 7184 . . . . . . . . . . . . . 14 (( bday Fn No ∧ 𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ ( bday β€œ 𝐡))
5234, 51mp3an1 1449 . . . . . . . . . . . . 13 ((𝐡 βŠ† No ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ ( bday β€œ 𝐡))
5352ad4ant14 751 . . . . . . . . . . . 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 3946 . . . . . . . . . . 11 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ ( bday β€˜π‘) ∈ 𝑂)
5549, 54eqeltrrd 2835 . . . . . . . . . 10 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ dom 𝑝 ∈ 𝑂)
56 ordelss 6334 . . . . . . . . . 10 ((Ord 𝑂 ∧ dom 𝑝 ∈ 𝑂) β†’ dom 𝑝 βŠ† 𝑂)
5745, 55, 56syl2anc 585 . . . . . . . . 9 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ dom 𝑝 βŠ† 𝑂)
5857sseld 3944 . . . . . . . 8 ((((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) ∧ 𝑝 ∈ 𝐡) β†’ (𝑧 ∈ dom 𝑝 β†’ 𝑧 ∈ 𝑂))
5958adantrd 493 . . . . . . 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 3149 . . . . . 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 4026 . . . . 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 483 . . . 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 3983 . . 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 811 . 2 (((𝐡 βŠ† No ∧ 𝐡 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday β€œ 𝐡) βŠ† 𝑂)) β†’ dom 𝑇 βŠ† 𝑂)
655, 64eqsstrd 3983 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070  βˆƒ!wreu 3350  βˆƒ*wrmo 3351   βˆͺ cun 3909   βŠ† wss 3911  ifcif 4487  {csn 4587  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634   β†Ύ cres 5636   β€œ cima 5637  Ord word 6317  Oncon0 6318  suc csuc 6320  β„©cio 6447   Fn wfn 6492  β€“ontoβ†’wfo 6495  β€˜cfv 6497  β„©crio 7313  1oc1o 8406   No csur 27004   <s cslt 27005   bday cbday 27006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noetalem1  27105
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