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Theorem nosupbnd1lem1 33282
Description: Lemma for nosupbnd1 33288. 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 1196 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝐴 No )
2 simp3 1135 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈𝐴)
31, 2sseldd 3943 . . 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 33277 . . . . 5 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
653ad2ant2 1131 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
7 nodmon 33231 . . . 4 (𝑆 No → dom 𝑆 ∈ On)
86, 7syl 17 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
9 noreson 33241 . . 3 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
103, 8, 9syl2anc 587 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
11 dmres 5853 . . . 4 dom (𝑈 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑈)
12 inss1 4179 . . . 4 (dom 𝑆 ∩ dom 𝑈) ⊆ dom 𝑆
1311, 12eqsstri 3976 . . 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 3965 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ⊆ dom 𝑆)
16 iffalse 4448 . . . . . . . . . . . 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 2869 . . . . . . . . . . 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 5751 . . . . . . . . . 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 6314 . . . . . . . . . . 11 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
20 eqid 2822 . . . . . . . . . . 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 6472 . . . . . . . . . 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 2873 . . . . . . . . 9 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
2322eleq2d 2899 . . . . . . . 8 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
24 vex 3472 . . . . . . . . 9 ∈ V
25 eleq1w 2896 . . . . . . . . . . . 12 (𝑦 = → (𝑦 ∈ dom 𝑢 ∈ dom 𝑢))
26 suceq 6234 . . . . . . . . . . . . . . . 16 (𝑦 = → suc 𝑦 = suc )
2726reseq2d 5831 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc ))
2826reseq2d 5831 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc ))
2927, 28eqeq12d 2838 . . . . . . . . . . . . . 14 (𝑦 = → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc ) = (𝑣 ↾ suc )))
3029imbi2d 344 . . . . . . . . . . . . 13 (𝑦 = → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3130ralbidv 3187 . . . . . . . . . . . 12 (𝑦 = → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3225, 31anbi12d 633 . . . . . . . . . . 11 (𝑦 = → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )))))
3332rexbidv 3283 . . . . . . . . . 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 5749 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
3534eleq2d 2899 . . . . . . . . . . . 12 (𝑢 = 𝑝 → ( ∈ dom 𝑢 ∈ dom 𝑝))
36 breq2 5046 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
3736notbid 321 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
38 reseq1 5825 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑢 ↾ suc ) = (𝑝 ↾ suc ))
3938eqeq1d 2824 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → ((𝑢 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑣 ↾ suc )))
4037, 39imbi12d 348 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4140ralbidv 3187 . . . . . . . . . . . 12 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4235, 41anbi12d 633 . . . . . . . . . . 11 (𝑢 = 𝑝 → (( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
4342cbvrexvw 3425 . . . . . . . . . 10 (∃𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4433, 43syl6bb 290 . . . . . . . . 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 3642 . . . . . . . 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 290 . . . . . . 7 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
47463ad2ant1 1130 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
48 simpl1 1188 . . . . . . . . 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 1189 . . . . . . . . 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 770 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝𝐴)
51 simprrl 780 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ dom 𝑝)
52 simprrr 781 . . . . . . . . 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 33281 . . . . . . . . 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 1379 . . . . . . . 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 1223 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝐴 No )
5655, 50sseldd 3943 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝 No )
573adantr 484 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈 No )
58 sltso 33255 . . . . . . . . . . . . . . 15 <s Or No
59 soasym 5481 . . . . . . . . . . . . . . 15 (( <s Or No ∧ (𝑝 No 𝑈 No )) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6058, 59mpan 689 . . . . . . . . . . . . . 14 ((𝑝 No 𝑈 No ) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6156, 57, 60syl2anc 587 . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈𝐴)
63 breq1 5045 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 <s 𝑝𝑈 <s 𝑝))
6463notbid 321 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝))
65 reseq1 5825 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 ↾ suc ) = (𝑈 ↾ suc ))
6665eqeq2d 2833 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → ((𝑝 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑈 ↾ suc )))
6764, 66imbi12d 348 . . . . . . . . . . . . . . 15 (𝑣 = 𝑈 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ) = (𝑈 ↾ suc ))))
6867rspcv 3593 . . . . . . . . . . . . . 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 410 . . . . . . . . . . 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 33231 . . . . . . . . . . . . . . . . 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 6194 . . . . . . . . . . . . . . . 16 ((dom 𝑝 ∈ On ∧ ∈ dom 𝑝) → ∈ On)
7573, 51, 74syl2anc 587 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ On)
76 sucelon 7517 . . . . . . . . . . . . . . 15 ( ∈ On ↔ suc ∈ On)
7775, 76sylib 221 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → suc ∈ On)
78 noreson 33241 . . . . . . . . . . . . . 14 ((𝑈 No ∧ suc ∈ On) → (𝑈 ↾ suc ) ∈ No )
7957, 77, 78syl2anc 587 . . . . . . . . . . . . 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 5473 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑈 ↾ suc ) ∈ No ) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8158, 80mpan 689 . . . . . . . . . . . . 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 484 . . . . . . . . . . 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 5060 . . . . . . . . . 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 416 . . . . . . . . 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 33243 . . . . . . . . . . 11 ((𝑝 No 𝑈 No ∧ suc ∈ On) → ((𝑝 ↾ suc ) <s (𝑈 ↾ suc ) → 𝑝 <s 𝑈))
8756, 57, 77, 86syl3anc 1368 . . . . . . . . . 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 155 . . . . . . . . 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 182 . . . . . . . 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 5060 . . . . . . 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 3271 . . . . . 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 243 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc )))
9392imp 410 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc ))
94 nodmord 33234 . . . . . . . 8 (𝑆 No → Ord dom 𝑆)
95 ordsucss 7518 . . . . . . . 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 410 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → suc ⊆ dom 𝑆)
9897resabs1d 5862 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ((𝑈 ↾ dom 𝑆) ↾ suc ) = (𝑈 ↾ suc ))
9998breq2d 5054 . . . 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 328 . . 3 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
101100ralrimiva 3174 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ∀ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
102 noresle 33274 . 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 1374 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  {cab 2800  wral 3130  wrex 3131  Vcvv 3469  cun 3906  cin 3907  wss 3908  ifcif 4439  {csn 4539  cop 4545   class class class wbr 5042  cmpt 5122   Or wor 5450  dom cdm 5532  cres 5534  Ord word 6168  Oncon0 6169  suc csuc 6171  cio 6291  cfv 6334  crio 7097  2oc2o 8083   No csur 33221   <s cslt 33222
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-1o 8089  df-2o 8090  df-no 33224  df-slt 33225  df-bday 33226
This theorem is referenced by:  nosupbnd1lem2  33283  nosupbnd1lem6  33287
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