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Theorem nosupbday 27750
Description: Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.)
Hypothesis
Ref Expression
nosupbday.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupbday (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → ( bday 𝑆) ⊆ 𝑂)
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑢,𝑂,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑂(𝑥,𝑣,𝑔)

Proof of Theorem nosupbday
StepHypRef Expression
1 nosupbday.1 . . . . 5 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
21nosupno 27748 . . . 4 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
32adantr 480 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → 𝑆 No )
4 bdayval 27693 . . 3 (𝑆 No → ( bday 𝑆) = dom 𝑆)
53, 4syl 17 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → ( bday 𝑆) = dom 𝑆)
6 iftrue 4531 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
71, 6eqtrid 2789 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
87dmeqd 5916 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
9 2oex 8517 . . . . . . . . 9 2o ∈ V
109dmsnop 6236 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
1110uneq2i 4165 . . . . . . 7 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
12 dmun 5921 . . . . . . 7 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩})
13 df-suc 6390 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
1411, 12, 133eqtr4i 2775 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
158, 14eqtrdi 2793 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1615adantr 480 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
17 simprrl 781 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → 𝑂 ∈ On)
18 eloni 6394 . . . . . 6 (𝑂 ∈ On → Ord 𝑂)
1917, 18syl 17 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → Ord 𝑂)
20 simprll 779 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → 𝐴 No )
21 simpl 482 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
22 nomaxmo 27743 . . . . . . . . . . . . 13 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2322adantr 480 . . . . . . . . . . . 12 ((𝐴 No 𝐴 ∈ V) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2423adantl 481 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
25 reu5 3382 . . . . . . . . . . 11 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
2621, 24, 25sylanbrc 583 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2726adantrr 717 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
28 riotacl 7405 . . . . . . . . 9 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
2927, 28syl 17 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
3020, 29sseldd 3984 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
31 bdayval 27693 . . . . . . 7 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
3230, 31syl 17 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
33 simprrr 782 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday 𝐴) ⊆ 𝑂)
34 bdayfo 27722 . . . . . . . . 9 bday : No onto→On
35 fofn 6822 . . . . . . . . 9 ( bday : No onto→On → bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7253 . . . . . . . 8 (( bday Fn No 𝐴 No ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3836, 20, 29, 37mp3an2i 1468 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3933, 38sseldd 3984 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ 𝑂)
4032, 39eqeltrrd 2842 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝑂)
41 ordsucss 7838 . . . . 5 (Ord 𝑂 → (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝑂 → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ 𝑂))
4219, 40, 41sylc 65 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ 𝑂)
4316, 42eqsstrd 4018 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆𝑂)
44 iffalse 4534 . . . . . . . 8 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
451, 44eqtrid 2789 . . . . . . 7 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
4645dmeqd 5916 . . . . . 6 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
47 iotaex 6534 . . . . . . 7 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
48 eqid 2737 . . . . . . 7 (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))
4947, 48dmmpti 6712 . . . . . 6 dom (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
5046, 49eqtrdi 2793 . . . . 5 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
5150adantr 480 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
52 simplrl 777 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → 𝑂 ∈ On)
53 ssel2 3978 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → 𝑢 No )
5453ad4ant14 752 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → 𝑢 No )
55 bdayval 27693 . . . . . . . . . . . 12 (𝑢 No → ( bday 𝑢) = dom 𝑢)
5654, 55syl 17 . . . . . . . . . . 11 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝑢) = dom 𝑢)
57 simplrr 778 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝐴) ⊆ 𝑂)
58 fnfvima 7253 . . . . . . . . . . . . . 14 (( bday Fn No 𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
5936, 58mp3an1 1450 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6059ad4ant14 752 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6157, 60sseldd 3984 . . . . . . . . . . 11 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝑢) ∈ 𝑂)
6256, 61eqeltrrd 2842 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → dom 𝑢𝑂)
63 onelss 6426 . . . . . . . . . 10 (𝑂 ∈ On → (dom 𝑢𝑂 → dom 𝑢𝑂))
6452, 62, 63sylc 65 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → dom 𝑢𝑂)
6564sseld 3982 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → (𝑦 ∈ dom 𝑢𝑦𝑂))
6665adantrd 491 . . . . . . 7 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦𝑂))
6766rexlimdva 3155 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦𝑂))
6867abssdv 4068 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ 𝑂)
6968adantl 481 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ 𝑂)
7051, 69eqsstrd 4018 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆𝑂)
7143, 70pm2.61ian 812 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → dom 𝑆𝑂)
725, 71eqsstrd 4018 1 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → ( bday 𝑆) ⊆ 𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  ∃!wreu 3378  ∃*wrmo 3379  Vcvv 3480  cun 3949  wss 3951  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225  dom cdm 5685  cres 5687  cima 5688  Ord word 6383  Oncon0 6384  suc csuc 6386  cio 6512   Fn wfn 6556  ontowfo 6559  cfv 6561  crio 7387  2oc2o 8500   No csur 27684   <s cslt 27685   bday cbday 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689
This theorem is referenced by:  noetalem1  27786
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