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Theorem nosupbday 26951
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 26949 . . . 4 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
32adantr 481 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → 𝑆 No )
4 bdayval 26894 . . 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 4478 . . . . . . . 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 2788 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
87dmeqd 5841 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
9 2oex 8370 . . . . . . . . 9 2o ∈ V
109dmsnop 6148 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
1110uneq2i 4106 . . . . . . 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 5846 . . . . . . 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 6302 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
1411, 12, 133eqtr4i 2774 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
158, 14eqtrdi 2792 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1615adantr 481 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
17 simprrl 778 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → 𝑂 ∈ On)
18 eloni 6306 . . . . . 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 776 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → 𝐴 No )
21 simpl 483 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
22 nomaxmo 26944 . . . . . . . . . . . . 13 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2322adantr 481 . . . . . . . . . . . 12 ((𝐴 No 𝐴 ∈ V) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2423adantl 482 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
25 reu5 3351 . . . . . . . . . . 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 714 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
28 riotacl 7304 . . . . . . . . 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 3932 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
31 bdayval 26894 . . . . . . 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 779 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday 𝐴) ⊆ 𝑂)
34 bdayfo 26923 . . . . . . . . 9 bday : No onto→On
35 fofn 6735 . . . . . . . . 9 ( bday : No onto→On → bday Fn No )
3634, 35ax-mp 5 . . . . . . . 8 bday Fn No
37 fnfvima 7159 . . . . . . . 8 (( bday Fn No 𝐴 No ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3836, 20, 29, 37mp3an2i 1465 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3933, 38sseldd 3932 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ 𝑂)
4032, 39eqeltrrd 2838 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝑂)
41 ordsucss 7723 . . . . 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 3969 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆𝑂)
44 iffalse 4481 . . . . . . . 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 2788 . . . . . . 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 5841 . . . . . 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 6446 . . . . . . 7 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
48 eqid 2736 . . . . . . 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 6622 . . . . . 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 2792 . . . . 5 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
5150adantr 481 . . . 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 774 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → 𝑂 ∈ On)
53 ssel2 3926 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → 𝑢 No )
5453ad4ant14 749 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → 𝑢 No )
55 bdayval 26894 . . . . . . . . . . . 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 775 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝐴) ⊆ 𝑂)
58 fnfvima 7159 . . . . . . . . . . . . . 14 (( bday Fn No 𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
5936, 58mp3an1 1447 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6059ad4ant14 749 . . . . . . . . . . . 12 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6157, 60sseldd 3932 . . . . . . . . . . 11 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → ( bday 𝑢) ∈ 𝑂)
6256, 61eqeltrrd 2838 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → dom 𝑢𝑂)
63 onelss 6338 . . . . . . . . . 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 3930 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) ∧ 𝑢𝐴) → (𝑦 ∈ dom 𝑢𝑦𝑂))
6665adantrd 492 . . . . . . 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 3148 . . . . . 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 4012 . . . . 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 482 . . . 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 3969 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂))) → dom 𝑆𝑂)
7143, 70pm2.61ian 809 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday 𝐴) ⊆ 𝑂)) → dom 𝑆𝑂)
725, 71eqsstrd 3969 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 396  w3a 1086   = wceq 1540  wcel 2105  {cab 2713  wral 3061  wrex 3070  ∃!wreu 3347  ∃*wrmo 3348  Vcvv 3441  cun 3895  wss 3897  ifcif 4472  {csn 4572  cop 4578   class class class wbr 5089  cmpt 5172  dom cdm 5614  cres 5616  cima 5617  Ord word 6295  Oncon0 6296  suc csuc 6298  cio 6423   Fn wfn 6468  ontowfo 6471  cfv 6473  crio 7285  2oc2o 8353   No csur 26886   <s cslt 26887   bday cbday 26888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6299  df-on 6300  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-1o 8359  df-2o 8360  df-no 26889  df-slt 26890  df-bday 26891
This theorem is referenced by:  noetalem1  26987
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