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Theorem nosupbday 33319
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) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
Distinct variable group:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem nosupbday
StepHypRef Expression
1 nosupbday.1 . . . 4 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 33317 . . 3 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
3 bdayval 33269 . . 3 (𝑆 No → ( bday 𝑆) = dom 𝑆)
42, 3syl 17 . 2 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) = dom 𝑆)
5 iftrue 4434 . . . . . . . 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⟩}))
61, 5syl5eq 2848 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
76dmeqd 5742 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8 2on 8098 . . . . . . . . . 10 2o ∈ On
98elexi 3463 . . . . . . . . 9 2o ∈ V
109dmsnop 6044 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
1110uneq2i 4090 . . . . . . 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 5747 . . . . . . 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 6169 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
1411, 12, 133eqtr4i 2834 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
157, 14eqtrdi 2852 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1615adantr 484 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
17 simprl 770 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → 𝐴 No )
18 simpl 486 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
19 nomaxmo 33315 . . . . . . . . . . . . 13 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2019adantr 484 . . . . . . . . . . . 12 ((𝐴 No 𝐴 ∈ V) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
2120adantl 485 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
22 reu5 3378 . . . . . . . . . . 11 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
2318, 21, 22sylanbrc 586 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
24 riotacl 7114 . . . . . . . . . 10 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
2523, 24syl 17 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
2617, 25sseldd 3919 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
27 bdayval 33269 . . . . . . . 8 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
2826, 27syl 17 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
29 bdayfo 33296 . . . . . . . . 9 bday : No onto→On
30 fofn 6571 . . . . . . . . 9 ( bday : No onto→On → bday Fn No )
3129, 30ax-mp 5 . . . . . . . 8 bday Fn No
32 fnfvima 6977 . . . . . . . 8 (( bday Fn No 𝐴 No ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3331, 17, 25, 32mp3an2i 1463 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → ( bday ‘(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ ( bday 𝐴))
3428, 33eqeltrrd 2894 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ ( bday 𝐴))
35 elssuni 4833 . . . . . 6 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ ( bday 𝐴) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ ( bday 𝐴))
3634, 35syl 17 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ ( bday 𝐴))
37 nodmord 33274 . . . . . . 7 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Ord dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
3826, 37syl 17 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → Ord dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
39 imassrn 5911 . . . . . . . 8 ( bday 𝐴) ⊆ ran bday
40 forn 6572 . . . . . . . . 9 ( bday : No onto→On → ran bday = On)
4129, 40ax-mp 5 . . . . . . . 8 ran bday = On
4239, 41sseqtri 3954 . . . . . . 7 ( bday 𝐴) ⊆ On
43 ssorduni 7484 . . . . . . 7 (( bday 𝐴) ⊆ On → Ord ( bday 𝐴))
4442, 43ax-mp 5 . . . . . 6 Ord ( bday 𝐴)
45 ordsucsssuc 7522 . . . . . 6 ((Ord dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ Ord ( bday 𝐴)) → (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ ( bday 𝐴) ↔ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc ( bday 𝐴)))
4638, 44, 45sylancl 589 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ ( bday 𝐴) ↔ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc ( bday 𝐴)))
4736, 46mpbid 235 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc ( bday 𝐴))
4816, 47eqsstrd 3956 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom 𝑆 ⊆ suc ( bday 𝐴))
49 iffalse 4437 . . . . . . . 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))))
501, 49syl5eq 2848 . . . . . . 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))))
5150dmeqd 5742 . . . . . 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))))
52 iotaex 6308 . . . . . . 7 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
53 eqid 2801 . . . . . . 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)))
5452, 53dmmpti 6468 . . . . . 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)))}
5551, 54eqtrdi 2852 . . . . 5 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
5655adantr 484 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
57 ssel2 3913 . . . . . . . . . . . . . 14 ((𝐴 No 𝑢𝐴) → 𝑢 No )
58 bdayval 33269 . . . . . . . . . . . . . 14 (𝑢 No → ( bday 𝑢) = dom 𝑢)
5957, 58syl 17 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) = dom 𝑢)
60 fnfvima 6977 . . . . . . . . . . . . . 14 (( bday Fn No 𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6131, 60mp3an1 1445 . . . . . . . . . . . . 13 ((𝐴 No 𝑢𝐴) → ( bday 𝑢) ∈ ( bday 𝐴))
6259, 61eqeltrrd 2894 . . . . . . . . . . . 12 ((𝐴 No 𝑢𝐴) → dom 𝑢 ∈ ( bday 𝐴))
6362adantlr 714 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑢𝐴) → dom 𝑢 ∈ ( bday 𝐴))
64 elssuni 4833 . . . . . . . . . . 11 (dom 𝑢 ∈ ( bday 𝐴) → dom 𝑢 ( bday 𝐴))
6563, 64syl 17 . . . . . . . . . 10 (((𝐴 No 𝐴 ∈ V) ∧ 𝑢𝐴) → dom 𝑢 ( bday 𝐴))
66 sssucid 6240 . . . . . . . . . 10 ( bday 𝐴) ⊆ suc ( bday 𝐴)
6765, 66sstrdi 3930 . . . . . . . . 9 (((𝐴 No 𝐴 ∈ V) ∧ 𝑢𝐴) → dom 𝑢 ⊆ suc ( bday 𝐴))
6867sseld 3917 . . . . . . . 8 (((𝐴 No 𝐴 ∈ V) ∧ 𝑢𝐴) → (𝑦 ∈ dom 𝑢𝑦 ∈ suc ( bday 𝐴)))
6968adantrd 495 . . . . . . 7 (((𝐴 No 𝐴 ∈ V) ∧ 𝑢𝐴) → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ suc ( bday 𝐴)))
7069rexlimdva 3246 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ suc ( bday 𝐴)))
7170abssdv 3999 . . . . 5 ((𝐴 No 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ suc ( bday 𝐴))
7271adantl 485 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ suc ( bday 𝐴))
7356, 72eqsstrd 3956 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V)) → dom 𝑆 ⊆ suc ( bday 𝐴))
7448, 73pm2.61ian 811 . 2 ((𝐴 No 𝐴 ∈ V) → dom 𝑆 ⊆ suc ( bday 𝐴))
754, 74eqsstrd 3956 1 ((𝐴 No 𝐴 ∈ V) → ( bday 𝑆) ⊆ suc ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  ∃!wreu 3111  ∃*wrmo 3112  Vcvv 3444  cun 3882  wss 3884  ifcif 4428  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  Ord word 6162  Oncon0 6163  suc csuc 6165  cio 6285   Fn wfn 6323  ontowfo 6326  cfv 6328  crio 7096  2oc2o 8083   No csur 33261   <s cslt 33262   bday cbday 33263
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-rep 5157  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-reu 3116  df-rmo 3117  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-iun 4886  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-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-1o 8089  df-2o 8090  df-no 33264  df-slt 33265  df-bday 33266
This theorem is referenced by:  noetalem4  33334
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