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Theorem noinfbday 27765
Description: Birthday bounding law for surreal infimum. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
noinfbday.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbday (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → ( bday 𝑇) ⊆ 𝑂)
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑔,𝑉
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑂(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfbday
Dummy variables 𝑝 𝑧 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noinfbday.1 . . . . 5 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
21noinfno 27763 . . . 4 ((𝐵 No 𝐵𝑉) → 𝑇 No )
3 bdayval 27693 . . . 4 (𝑇 No → ( bday 𝑇) = dom 𝑇)
42, 3syl 17 . . 3 ((𝐵 No 𝐵𝑉) → ( bday 𝑇) = dom 𝑇)
54adantr 480 . 2 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → ( bday 𝑇) = dom 𝑇)
6 iftrue 4531 . . . . . . . 8 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
71, 6eqtrid 2789 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
87dmeqd 5916 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
9 1oex 8516 . . . . . . . . 9 1o ∈ V
109dmsnop 6236 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
1110uneq2i 4165 . . . . . . 7 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
12 dmun 5921 . . . . . . 7 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩})
13 df-suc 6390 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
1411, 12, 133eqtr4i 2775 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
158, 14eqtrdi 2793 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
1615adantr 480 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
17 simprrl 781 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → 𝑂 ∈ On)
18 eloni 6394 . . . . . 6 (𝑂 ∈ On → Ord 𝑂)
1917, 18syl 17 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → Ord 𝑂)
20 simprll 779 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → 𝐵 No )
21 simpl 482 . . . . . . . . . 10 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
22 nominmo 27744 . . . . . . . . . . 11 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2320, 22syl 17 . . . . . . . . . 10 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
24 reu5 3382 . . . . . . . . . 10 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
2521, 23, 24sylanbrc 583 . . . . . . . . 9 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
26 riotacl 7405 . . . . . . . . 9 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
2725, 26syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
2820, 27sseldd 3984 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
29 bdayval 27693 . . . . . . 7 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
3028, 29syl 17 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
31 simprrr 782 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday 𝐵) ⊆ 𝑂)
32 bdayfo 27722 . . . . . . . . 9 bday : No onto→On
33 fofn 6822 . . . . . . . . 9 ( bday : No onto→On → bday Fn No )
3432, 33ax-mp 5 . . . . . . . 8 bday Fn No
35 fnfvima 7253 . . . . . . . 8 (( bday Fn No 𝐵 No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ ( bday 𝐵))
3634, 20, 27, 35mp3an2i 1468 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ ( bday 𝐵))
3731, 36sseldd 3984 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ 𝑂)
3830, 37eqeltrrd 2842 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝑂)
39 ordsucss 7838 . . . . 5 (Ord 𝑂 → (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝑂 → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ 𝑂))
4019, 38, 39sylc 65 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ 𝑂)
4116, 40eqsstrd 4018 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom 𝑇𝑂)
421noinfdm 27764 . . . . 5 (¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
4342adantr 480 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom 𝑇 = {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
44 simplrl 777 . . . . . . . . . . 11 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → 𝑂 ∈ On)
4544, 18syl 17 . . . . . . . . . 10 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → Ord 𝑂)
46 ssel2 3978 . . . . . . . . . . . . 13 ((𝐵 No 𝑝𝐵) → 𝑝 No )
4746ad4ant14 752 . . . . . . . . . . . 12 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → 𝑝 No )
48 bdayval 27693 . . . . . . . . . . . 12 (𝑝 No → ( bday 𝑝) = dom 𝑝)
4947, 48syl 17 . . . . . . . . . . 11 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ( bday 𝑝) = dom 𝑝)
50 simplrr 778 . . . . . . . . . . . 12 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ( bday 𝐵) ⊆ 𝑂)
51 fnfvima 7253 . . . . . . . . . . . . . 14 (( bday Fn No 𝐵 No 𝑝𝐵) → ( bday 𝑝) ∈ ( bday 𝐵))
5234, 51mp3an1 1450 . . . . . . . . . . . . 13 ((𝐵 No 𝑝𝐵) → ( bday 𝑝) ∈ ( bday 𝐵))
5352ad4ant14 752 . . . . . . . . . . . 12 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ( bday 𝑝) ∈ ( bday 𝐵))
5450, 53sseldd 3984 . . . . . . . . . . 11 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ( bday 𝑝) ∈ 𝑂)
5549, 54eqeltrrd 2842 . . . . . . . . . 10 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → dom 𝑝𝑂)
56 ordelss 6400 . . . . . . . . . 10 ((Ord 𝑂 ∧ dom 𝑝𝑂) → dom 𝑝𝑂)
5745, 55, 56syl2anc 584 . . . . . . . . 9 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → dom 𝑝𝑂)
5857sseld 3982 . . . . . . . 8 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → (𝑧 ∈ dom 𝑝𝑧𝑂))
5958adantrd 491 . . . . . . 7 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ((𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) → 𝑧𝑂))
6059rexlimdva 3155 . . . . . 6 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → (∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) → 𝑧𝑂))
6160abssdv 4068 . . . . 5 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ⊆ 𝑂)
6261adantl 481 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → {𝑧 ∣ ∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ⊆ 𝑂)
6343, 62eqsstrd 4018 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom 𝑇𝑂)
6441, 63pm2.61ian 812 . 2 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → dom 𝑇𝑂)
655, 64eqsstrd 4018 1 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → ( bday 𝑇) ⊆ 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  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  1oc1o 8499   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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