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Theorem noinfbday 27780
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   𝑥,𝑣,𝑦
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 𝑞 𝑧 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 27778 . . . 4 ((𝐵 No 𝐵𝑉) → 𝑇 No )
3 bdayval 27708 . . . 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 4537 . . . . . . . 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 2787 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
87dmeqd 5919 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
9 1oex 8515 . . . . . . . . 9 1o ∈ V
109dmsnop 6238 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
1110uneq2i 4175 . . . . . . 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 5924 . . . . . . 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 6392 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
1411, 12, 133eqtr4i 2773 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
158, 14eqtrdi 2791 . . . . 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 6396 . . . . . 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 27759 . . . . . . . . . . 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 3380 . . . . . . . . . 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 3996 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
29 bdayval 27708 . . . . . . 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 27737 . . . . . . . . 9 bday : No onto→On
33 fofn 6823 . . . . . . . . 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 1465 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ ( bday 𝐵))
3731, 36sseldd 3996 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → ( bday ‘(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ 𝑂)
3830, 37eqeltrrd 2840 . . . . 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 4034 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂))) → dom 𝑇𝑂)
421noinfdm 27779 . . . . 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 3990 . . . . . . . . . . . . 13 ((𝐵 No 𝑝𝐵) → 𝑝 No )
4746ad4ant14 752 . . . . . . . . . . . 12 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → 𝑝 No )
48 bdayval 27708 . . . . . . . . . . . 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 1447 . . . . . . . . . . . . 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 3996 . . . . . . . . . . 11 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → ( bday 𝑝) ∈ 𝑂)
5549, 54eqeltrrd 2840 . . . . . . . . . 10 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → dom 𝑝𝑂)
56 ordelss 6402 . . . . . . . . . 10 ((Ord 𝑂 ∧ dom 𝑝𝑂) → dom 𝑝𝑂)
5745, 55, 56syl2anc 584 . . . . . . . . 9 ((((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) ∧ 𝑝𝐵) → dom 𝑝𝑂)
5857sseld 3994 . . . . . . . 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 3153 . . . . . 6 (((𝐵 No 𝐵𝑉) ∧ (𝑂 ∈ On ∧ ( bday 𝐵) ⊆ 𝑂)) → (∃𝑝𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞𝐵𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) → 𝑧𝑂))
6160abssdv 4078 . . . . 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 4034 . . 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 4034 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 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  cun 3961  wss 3963  ifcif 4531  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231  dom cdm 5689  cres 5691  cima 5692  Ord word 6385  Oncon0 6386  suc csuc 6388  cio 6514   Fn wfn 6558  ontowfo 6561  cfv 6563  crio 7387  1oc1o 8498   No csur 27699   <s cslt 27700   bday cbday 27701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704
This theorem is referenced by:  noetalem1  27801
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