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Theorem nobnddown 30903
Description: Any set of surreals is bounded below by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
Assertion
Ref Expression
nobnddown ((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem nobnddown
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3181 . 2 (𝐴𝑉𝐴 ∈ V)
2 1on 7428 . . . . . . 7 1𝑜 ∈ On
32elexi 3182 . . . . . 6 1𝑜 ∈ V
43prid1 4237 . . . . 5 1𝑜 ∈ {1𝑜, 2𝑜}
5 eqid 2606 . . . . 5 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} = {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜}
64, 5nobndlem2 30895 . . . 4 ((𝐴 No 𝐴 ∈ V) → {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On)
7 noxp1o 30866 . . . 4 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
86, 7syl 17 . . 3 ((𝐴 No 𝐴 ∈ V) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
98adantr 479 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
10 ssel2 3559 . . . . . 6 ((𝐴 No 𝑦𝐴) → 𝑦 No )
1110adantlr 746 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → 𝑦 No )
124nobndlem4 30897 . . . . . . . 8 (𝑦 No {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
1310, 12syl 17 . . . . . . 7 ((𝐴 No 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
1413adantlr 746 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
156adantr 479 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On)
164, 5nobndlem6 30899 . . . . . . . . . . . 12 ((𝐴 No 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
1716adantlr 746 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
18 onelss 5666 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜}))
1915, 17, 18sylc 62 . . . . . . . . . 10 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
2019sselda 3564 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → 𝑑 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
213fvconst2 6349 . . . . . . . . 9 (𝑑 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = 1𝑜)
2220, 21syl 17 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = 1𝑜)
2314adantr 479 . . . . . . . . . . . . . . . 16 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
24 onss 6856 . . . . . . . . . . . . . . . . . 18 ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ On)
2514, 24syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ On)
2625sselda 3564 . . . . . . . . . . . . . . . 16 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → 𝑑 ∈ On)
27 ontri1 5657 . . . . . . . . . . . . . . . 16 (( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On ∧ 𝑑 ∈ On) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 ↔ ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
2823, 26, 27syl2anc 690 . . . . . . . . . . . . . . 15 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 ↔ ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
2928biimpd 217 . . . . . . . . . . . . . 14 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 → ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
3029con2d 127 . . . . . . . . . . . . 13 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑))
3130ex 448 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)))
3231pm2.43d 50 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑))
3332imp 443 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)
34 intss1 4418 . . . . . . . . . 10 (𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)
3533, 34nsyl 133 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ¬ 𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜})
36 df-ne 2778 . . . . . . . . . 10 ((𝑦𝑑) ≠ 1𝑜 ↔ ¬ (𝑦𝑑) = 1𝑜)
37 fveq2 6085 . . . . . . . . . . . . . 14 (𝑘 = 𝑑 → (𝑦𝑘) = (𝑦𝑑))
3837neeq1d 2837 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → ((𝑦𝑘) ≠ 1𝑜 ↔ (𝑦𝑑) ≠ 1𝑜))
3938elrab3 3328 . . . . . . . . . . . 12 (𝑑 ∈ On → (𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ↔ (𝑦𝑑) ≠ 1𝑜))
4039biimprd 236 . . . . . . . . . . 11 (𝑑 ∈ On → ((𝑦𝑑) ≠ 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4126, 40syl 17 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ((𝑦𝑑) ≠ 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4236, 41syl5bir 231 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (¬ (𝑦𝑑) = 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4335, 42mt3d 138 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (𝑦𝑑) = 1𝑜)
4422, 43eqtr4d 2643 . . . . . . 7 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑))
4544ralrimiva 2945 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑))
463fvconst2 6349 . . . . . . . 8 ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
4717, 46syl 17 . . . . . . 7 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
484nobndlem5 30898 . . . . . . . . . . . . . 14 (𝑦 No → (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ≠ 1𝑜)
4911, 48syl 17 . . . . . . . . . . . . 13 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ≠ 1𝑜)
5049neneqd 2783 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ¬ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
51 nofv 30857 . . . . . . . . . . . . 13 (𝑦 No → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
5211, 51syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
53 3orel2 30650 . . . . . . . . . . . 12 (¬ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 → (((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5450, 52, 53sylc 62 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
55 eqid 2606 . . . . . . . . . . 11 1𝑜 = 1𝑜
5654, 55jctil 557 . . . . . . . . . 10 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (1𝑜 = 1𝑜 ∧ ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
57 andi 906 . . . . . . . . . 10 ((1𝑜 = 1𝑜 ∧ ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ↔ ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5856, 57sylib 206 . . . . . . . . 9 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5958orcd 405 . . . . . . . 8 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
60 fvex 6095 . . . . . . . . . 10 (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ∈ V
613, 60brtp 30695 . . . . . . . . 9 (1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ↔ ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
62 df-3or 1031 . . . . . . . . 9 (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ↔ (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
6361, 62bitri 262 . . . . . . . 8 (1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ↔ (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
6459, 63sylibr 222 . . . . . . 7 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → 1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
6547, 64eqbrtrd 4596 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
66 raleq 3111 . . . . . . . 8 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ↔ ∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑)))
67 fveq2 6085 . . . . . . . . 9 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐) = (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
68 fveq2 6085 . . . . . . . . 9 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (𝑦𝑐) = (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
6967, 68breq12d 4587 . . . . . . . 8 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐) ↔ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜})))
7066, 69anbi12d 742 . . . . . . 7 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ((∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)) ↔ (∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))))
7170rspcev 3278 . . . . . 6 (( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On ∧ (∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))) → ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)))
7214, 45, 65, 71syl12anc 1315 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)))
73 sltval 30847 . . . . . 6 ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No 𝑦 No ) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ↔ ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐))))
7473biimpar 500 . . . . 5 (((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No 𝑦 No ) ∧ ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐))) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
759, 11, 72, 74syl21anc 1316 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
7675ralrimiva 2945 . . 3 ((𝐴 No 𝐴 ∈ V) → ∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
774, 5nobndlem8 30901 . . 3 ((𝐴 No 𝐴 ∈ V) → ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))
78 breq1 4577 . . . . . 6 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (𝑥 <s 𝑦 ↔ ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦))
7978ralbidv 2965 . . . . 5 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (∀𝑦𝐴 𝑥 <s 𝑦 ↔ ∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦))
80 fveq2 6085 . . . . . 6 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → ( bday 𝑥) = ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})))
8180sseq1d 3591 . . . . 5 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (( bday 𝑥) ⊆ suc ( bday 𝐴) ↔ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴)))
8279, 81anbi12d 742 . . . 4 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → ((∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)) ↔ (∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ∧ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))))
8382rspcev 3278 . . 3 ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No ∧ (∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ∧ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
848, 76, 77, 83syl12anc 1315 . 2 ((𝐴 No 𝐴 ∈ V) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
851, 84sylan2 489 1 ((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3o 1029   = wceq 1474  wcel 1976  wne 2776  wral 2892  wrex 2893  {crab 2896  Vcvv 3169  wss 3536  c0 3870  {csn 4121  {ctp 4125  cop 4127   cuni 4363   cint 4401   class class class wbr 4574   × cxp 5023  cima 5028  Oncon0 5623  suc csuc 5625  cfv 5787  1𝑜c1o 7414  2𝑜c2o 7415   No csur 30840   <s cslt 30841   bday cbday 30842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-ord 5626  df-on 5627  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-1o 7421  df-2o 7422  df-no 30843  df-slt 30844  df-bday 30845
This theorem is referenced by:  nofulllem2  30905
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