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Theorem nosupbnd1lem1 32180
Description: Lemma for nosupbnd1 32186. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.)
Hypothesis
Ref Expression
nosupbnd1.1 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
nosupbnd1lem1 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦,𝑣
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑥,𝑦,𝑢,𝑔)

Proof of Theorem nosupbnd1lem1
Dummy variables 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2l 1249 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝐴 No )
2 simp3 1161 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈𝐴)
31, 2sseldd 3806 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈 No )
4 nosupbnd1.1 . . . . . 6 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))))
54nosupno 32175 . . . . 5 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
653ad2ant2 1157 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
7 nodmon 32129 . . . 4 (𝑆 No → dom 𝑆 ∈ On)
86, 7syl 17 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
9 noreson 32139 . . 3 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
103, 8, 9syl2anc 575 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
11 dmres 5629 . . . 4 dom (𝑈 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑈)
12 inss1 4036 . . . 4 (dom 𝑆 ∩ dom 𝑈) ⊆ dom 𝑆
1311, 12eqsstri 3839 . . 3 dom (𝑈 ↾ dom 𝑆) ⊆ dom 𝑆
1413a1i 11 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom (𝑈 ↾ dom 𝑆) ⊆ dom 𝑆)
15 ssidd 3828 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ⊆ dom 𝑆)
16 iffalse 4295 . . . . . . . . . . . 12 (¬ ∃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))))
174, 16syl5eq 2859 . . . . . . . . . . 11 (¬ ∃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))))
1817dmeqd 5534 . . . . . . . . . 10 (¬ ∃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))))
19 iotaex 6084 . . . . . . . . . . 11 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
20 eqid 2813 . . . . . . . . . . 11 (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)))
2119, 20dmmpti 6237 . . . . . . . . . 10 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)))}
2218, 21syl6eq 2863 . . . . . . . . 9 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
2322eleq2d 2878 . . . . . . . 8 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
24 vex 3401 . . . . . . . . 9 ∈ V
25 eleq1w 2875 . . . . . . . . . . . 12 (𝑦 = → (𝑦 ∈ dom 𝑢 ∈ dom 𝑢))
26 suceq 6009 . . . . . . . . . . . . . . . 16 (𝑦 = → suc 𝑦 = suc )
2726reseq2d 5604 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc ))
2826reseq2d 5604 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc ))
2927, 28eqeq12d 2828 . . . . . . . . . . . . . 14 (𝑦 = → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc ) = (𝑣 ↾ suc )))
3029imbi2d 331 . . . . . . . . . . . . 13 (𝑦 = → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3130ralbidv 3181 . . . . . . . . . . . 12 (𝑦 = → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3225, 31anbi12d 618 . . . . . . . . . . 11 (𝑦 = → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )))))
3332rexbidv 3247 . . . . . . . . . 10 (𝑦 = → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )))))
34 dmeq 5532 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
3534eleq2d 2878 . . . . . . . . . . . 12 (𝑢 = 𝑝 → ( ∈ dom 𝑢 ∈ dom 𝑝))
36 breq2 4855 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
3736notbid 309 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
38 reseq1 5598 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑢 ↾ suc ) = (𝑝 ↾ suc ))
3938eqeq1d 2815 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → ((𝑢 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑣 ↾ suc )))
4037, 39imbi12d 335 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4140ralbidv 3181 . . . . . . . . . . . 12 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4235, 41anbi12d 618 . . . . . . . . . . 11 (𝑢 = 𝑝 → (( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
4342cbvrexv 3368 . . . . . . . . . 10 (∃𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4433, 43syl6bb 278 . . . . . . . . 9 (𝑦 = → (∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
4524, 44elab 3552 . . . . . . . 8 ( ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4623, 45syl6bb 278 . . . . . . 7 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
47463ad2ant1 1156 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
48 simpl1 1235 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
49 simpl2 1237 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝐴 No 𝐴 ∈ V))
50 simprl 778 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝𝐴)
51 simprrl 790 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ dom 𝑝)
52 simprrr 791 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))
534nosupres 32179 . . . . . . . . 9 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑝𝐴 ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))) → (𝑆 ↾ suc ) = (𝑝 ↾ suc ))
5448, 49, 50, 51, 52, 53syl113anc 1494 . . . . . . . 8 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝑆 ↾ suc ) = (𝑝 ↾ suc ))
55 simpl2l 1290 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝐴 No )
5655, 50sseldd 3806 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝 No )
573adantr 468 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈 No )
58 sltso 32153 . . . . . . . . . . . . . . 15 <s Or No
59 soasym 31984 . . . . . . . . . . . . . . 15 (( <s Or No ∧ (𝑝 No 𝑈 No )) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6058, 59mpan 673 . . . . . . . . . . . . . 14 ((𝑝 No 𝑈 No ) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6156, 57, 60syl2anc 575 . . . . . . . . . . . . 13 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
62 simpl3 1239 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈𝐴)
63 breq1 4854 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 <s 𝑝𝑈 <s 𝑝))
6463notbid 309 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝))
65 reseq1 5598 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 ↾ suc ) = (𝑈 ↾ suc ))
6665eqeq2d 2823 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → ((𝑝 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑈 ↾ suc )))
6764, 66imbi12d 335 . . . . . . . . . . . . . . 15 (𝑣 = 𝑈 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ) = (𝑈 ↾ suc ))))
6867rspcv 3505 . . . . . . . . . . . . . 14 (𝑈𝐴 → (∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )) → (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ) = (𝑈 ↾ suc ))))
6962, 52, 68sylc 65 . . . . . . . . . . . . 13 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ) = (𝑈 ↾ suc )))
7061, 69syld 47 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝑝 <s 𝑈 → (𝑝 ↾ suc ) = (𝑈 ↾ suc )))
7170imp 395 . . . . . . . . . . 11 ((((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) ∧ 𝑝 <s 𝑈) → (𝑝 ↾ suc ) = (𝑈 ↾ suc ))
72 nodmon 32129 . . . . . . . . . . . . . . . . 17 (𝑝 No → dom 𝑝 ∈ On)
7356, 72syl 17 . . . . . . . . . . . . . . . 16 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → dom 𝑝 ∈ On)
74 onelon 5968 . . . . . . . . . . . . . . . 16 ((dom 𝑝 ∈ On ∧ ∈ dom 𝑝) → ∈ On)
7573, 51, 74syl2anc 575 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ On)
76 sucelon 7250 . . . . . . . . . . . . . . 15 ( ∈ On ↔ suc ∈ On)
7775, 76sylib 209 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → suc ∈ On)
78 noreson 32139 . . . . . . . . . . . . . 14 ((𝑈 No ∧ suc ∈ On) → (𝑈 ↾ suc ) ∈ No )
7957, 77, 78syl2anc 575 . . . . . . . . . . . . 13 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝑈 ↾ suc ) ∈ No )
80 sonr 5260 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑈 ↾ suc ) ∈ No ) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8158, 80mpan 673 . . . . . . . . . . . . 13 ((𝑈 ↾ suc ) ∈ No → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8279, 81syl 17 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8382adantr 468 . . . . . . . . . . 11 ((((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) ∧ 𝑝 <s 𝑈) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8471, 83eqnbrtrd 4869 . . . . . . . . . 10 ((((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) ∧ 𝑝 <s 𝑈) → ¬ (𝑝 ↾ suc ) <s (𝑈 ↾ suc ))
8584ex 399 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (𝑝 <s 𝑈 → ¬ (𝑝 ↾ suc ) <s (𝑈 ↾ suc )))
86 sltres 32141 . . . . . . . . . . 11 ((𝑝 No 𝑈 No ∧ suc ∈ On) → ((𝑝 ↾ suc ) <s (𝑈 ↾ suc ) → 𝑝 <s 𝑈))
8756, 57, 77, 86syl3anc 1483 . . . . . . . . . 10 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ((𝑝 ↾ suc ) <s (𝑈 ↾ suc ) → 𝑝 <s 𝑈))
8887con3d 149 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → (¬ 𝑝 <s 𝑈 → ¬ (𝑝 ↾ suc ) <s (𝑈 ↾ suc )))
8985, 88pm2.61d 171 . . . . . . . 8 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ¬ (𝑝 ↾ suc ) <s (𝑈 ↾ suc ))
9054, 89eqnbrtrd 4869 . . . . . . 7 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc ))
9190rexlimdvaa 3227 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))) → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc )))
9247, 91sylbid 231 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc )))
9392imp 395 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc ))
94 nodmord 32132 . . . . . . . 8 (𝑆 No → Ord dom 𝑆)
95 ordsucss 7251 . . . . . . . 8 (Ord dom 𝑆 → ( ∈ dom 𝑆 → suc ⊆ dom 𝑆))
966, 94, 953syl 18 . . . . . . 7 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 → suc ⊆ dom 𝑆))
9796imp 395 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → suc ⊆ dom 𝑆)
9897resabs1d 5638 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ((𝑈 ↾ dom 𝑆) ↾ suc ) = (𝑈 ↾ suc ))
9998breq2d 4863 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ((𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ) ↔ (𝑆 ↾ suc ) <s (𝑈 ↾ suc )))
10093, 99mtbird 316 . . 3 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
101100ralrimiva 3161 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ∀ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
102 noresle 32172 . 2 ((((𝑈 ↾ dom 𝑆) ∈ No 𝑆 No ) ∧ (dom (𝑈 ↾ dom 𝑆) ⊆ dom 𝑆 ∧ dom 𝑆 ⊆ dom 𝑆 ∧ ∀ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
10310, 6, 14, 15, 101, 102syl23anc 1489 1 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  {cab 2799  wral 3103  wrex 3104  Vcvv 3398  cun 3774  cin 3775  wss 3776  ifcif 4286  {csn 4377  cop 4383   class class class wbr 4851  cmpt 4930   Or wor 5238  dom cdm 5318  cres 5320  Ord word 5942  Oncon0 5943  suc csuc 5945  cio 6065  cfv 6104  crio 6837  2𝑜c2o 7793   No csur 32119   <s cslt 32120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-ord 5946  df-on 5947  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-1o 7799  df-2o 7800  df-no 32122  df-slt 32123  df-bday 32124
This theorem is referenced by:  nosupbnd1lem2  32181  nosupbnd1lem6  32185
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