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Theorem nosupbnd1lem1 27553
Description: Lemma for nosupbnd1 27559. 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 1198 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝐴 No )
2 simp3 1137 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈𝐴)
31, 2sseldd 3983 . . 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 27548 . . . . 5 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
653ad2ant2 1133 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
7 nodmon 27495 . . . 4 (𝑆 No → dom 𝑆 ∈ On)
86, 7syl 17 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
9 noreson 27505 . . 3 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
103, 8, 9syl2anc 583 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
11 dmres 6003 . . . 4 dom (𝑈 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑈)
12 inss1 4228 . . . 4 (dom 𝑆 ∩ dom 𝑈) ⊆ dom 𝑆
1311, 12eqsstri 4016 . . 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 4005 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ⊆ dom 𝑆)
16 iffalse 4537 . . . . . . . . . . . 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, 16eqtrid 2783 . . . . . . . . . . 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 5905 . . . . . . . . . 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 6516 . . . . . . . . . . 11 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
20 eqid 2731 . . . . . . . . . . 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 6694 . . . . . . . . . 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, 21eqtrdi 2787 . . . . . . . . 9 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
2322eleq2d 2818 . . . . . . . 8 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
24 vex 3477 . . . . . . . . 9 ∈ V
25 eleq1w 2815 . . . . . . . . . . . 12 (𝑦 = → (𝑦 ∈ dom 𝑢 ∈ dom 𝑢))
26 suceq 6430 . . . . . . . . . . . . . . . 16 (𝑦 = → suc 𝑦 = suc )
2726reseq2d 5981 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc ))
2826reseq2d 5981 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc ))
2927, 28eqeq12d 2747 . . . . . . . . . . . . . 14 (𝑦 = → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc ) = (𝑣 ↾ suc )))
3029imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3130ralbidv 3176 . . . . . . . . . . . 12 (𝑦 = → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))))
3225, 31anbi12d 630 . . . . . . . . . . 11 (𝑦 = → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )))))
3332rexbidv 3177 . . . . . . . . . 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 5903 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
3534eleq2d 2818 . . . . . . . . . . . 12 (𝑢 = 𝑝 → ( ∈ dom 𝑢 ∈ dom 𝑝))
36 breq2 5152 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑣 <s 𝑢𝑣 <s 𝑝))
3736notbid 318 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
38 reseq1 5975 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑢 ↾ suc ) = (𝑝 ↾ suc ))
3938eqeq1d 2733 . . . . . . . . . . . . . 14 (𝑢 = 𝑝 → ((𝑢 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑣 ↾ suc )))
4037, 39imbi12d 344 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4140ralbidv 3176 . . . . . . . . . . . 12 (𝑢 = 𝑝 → (∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc )) ↔ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4235, 41anbi12d 630 . . . . . . . . . . 11 (𝑢 = 𝑝 → (( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
4342cbvrexvw 3234 . . . . . . . . . 10 (∃𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))
4433, 43bitrdi 287 . . . . . . . . 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 3668 . . . . . . . 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, 45bitrdi 287 . . . . . . 7 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
47463ad2ant1 1132 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 ↔ ∃𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
48 simpl1 1190 . . . . . . . . 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 1191 . . . . . . . . 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 768 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝𝐴)
51 simprrl 778 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ dom 𝑝)
52 simprrr 779 . . . . . . . . 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 27552 . . . . . . . . 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 1381 . . . . . . . 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 1225 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝐴 No )
5655, 50sseldd 3983 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝 No )
573adantr 480 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈 No )
58 sltso 27521 . . . . . . . . . . . . . . 15 <s Or No
59 soasym 5619 . . . . . . . . . . . . . . 15 (( <s Or No ∧ (𝑝 No 𝑈 No )) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6058, 59mpan 687 . . . . . . . . . . . . . 14 ((𝑝 No 𝑈 No ) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6156, 57, 60syl2anc 583 . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑈𝐴)
63 breq1 5151 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 <s 𝑝𝑈 <s 𝑝))
6463notbid 318 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝))
65 reseq1 5975 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 ↾ suc ) = (𝑈 ↾ suc ))
6665eqeq2d 2742 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑈 → ((𝑝 ↾ suc ) = (𝑣 ↾ suc ) ↔ (𝑝 ↾ suc ) = (𝑈 ↾ suc )))
6764, 66imbi12d 344 . . . . . . . . . . . . . . 15 (𝑣 = 𝑈 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )) ↔ (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ) = (𝑈 ↾ suc ))))
6867rspcv 3608 . . . . . . . . . . . . . 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 406 . . . . . . . . . . 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 27495 . . . . . . . . . . . . . . . . 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 6389 . . . . . . . . . . . . . . . 16 ((dom 𝑝 ∈ On ∧ ∈ dom 𝑝) → ∈ On)
7573, 51, 74syl2anc 583 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ On)
76 onsucb 7809 . . . . . . . . . . . . . . 15 ( ∈ On ↔ suc ∈ On)
7775, 76sylib 217 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → suc ∈ On)
78 noreson 27505 . . . . . . . . . . . . . 14 ((𝑈 No ∧ suc ∈ On) → (𝑈 ↾ suc ) ∈ No )
7957, 77, 78syl2anc 583 . . . . . . . . . . . . 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 5611 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑈 ↾ suc ) ∈ No ) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8158, 80mpan 687 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 5166 . . . . . . . . . 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 412 . . . . . . . . 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 27507 . . . . . . . . . . 11 ((𝑝 No 𝑈 No ∧ suc ∈ On) → ((𝑝 ↾ suc ) <s (𝑈 ↾ suc ) → 𝑝 <s 𝑈))
8756, 57, 77, 86syl3anc 1370 . . . . . . . . . 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 152 . . . . . . . . 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 179 . . . . . . . 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 5166 . . . . . . 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 3155 . . . . . 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 239 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ( ∈ dom 𝑆 → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc )))
9392imp 406 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s (𝑈 ↾ suc ))
94 nodmord 27498 . . . . . . . 8 (𝑆 No → Ord dom 𝑆)
95 ordsucss 7810 . . . . . . . 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 406 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → suc ⊆ dom 𝑆)
9897resabs1d 6012 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ((𝑈 ↾ dom 𝑆) ↾ suc ) = (𝑈 ↾ suc ))
9998breq2d 5160 . . . 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 325 . . 3 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
101100ralrimiva 3145 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ∀ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
102 noresle 27542 . 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 1376 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  Vcvv 3473  cun 3946  cin 3947  wss 3948  ifcif 4528  {csn 4628  cop 4634   class class class wbr 5148  cmpt 5231   Or wor 5587  dom cdm 5676  cres 5678  Ord word 6363  Oncon0 6364  suc csuc 6366  cio 6493  cfv 6543  crio 7367  2oc2o 8466   No csur 27485   <s cslt 27486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-1o 8472  df-2o 8473  df-no 27488  df-slt 27489  df-bday 27490
This theorem is referenced by:  nosupbnd1lem2  27554  nosupbnd1lem6  27558
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