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Theorem nosupbnd1lem1 27768
Description: Lemma for nosupbnd1 27774. 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 3996 . . 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 27763 . . . . 5 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
653ad2ant2 1133 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
7 nodmon 27710 . . . 4 (𝑆 No → dom 𝑆 ∈ On)
86, 7syl 17 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
9 noreson 27720 . . 3 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
103, 8, 9syl2anc 584 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
11 dmres 6032 . . . 4 dom (𝑈 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑈)
12 inss1 4245 . . . 4 (dom 𝑆 ∩ dom 𝑈) ⊆ dom 𝑆
1311, 12eqsstri 4030 . . 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 4019 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ⊆ dom 𝑆)
16 iffalse 4540 . . . . . . . . . . . 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 2787 . . . . . . . . . . 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 5919 . . . . . . . . . 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 6536 . . . . . . . . . . 11 (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)) ∈ V
20 eqid 2735 . . . . . . . . . . 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 6713 . . . . . . . . . 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 2791 . . . . . . . . 9 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
2322eleq2d 2825 . . . . . . . 8 (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
24 vex 3482 . . . . . . . . 9 ∈ V
25 eleq1w 2822 . . . . . . . . . . . 12 (𝑦 = → (𝑦 ∈ dom 𝑢 ∈ dom 𝑢))
26 suceq 6452 . . . . . . . . . . . . . . . 16 (𝑦 = → suc 𝑦 = suc )
2726reseq2d 6000 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc ))
2826reseq2d 6000 . . . . . . . . . . . . . . 15 (𝑦 = → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc ))
2927, 28eqeq12d 2751 . . . . . . . . . . . . . 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 632 . . . . . . . . . . 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 5917 . . . . . . . . . . . . 13 (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
3534eleq2d 2825 . . . . . . . . . . . 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 5994 . . . . . . . . . . . . . . 15 (𝑢 = 𝑝 → (𝑢 ↾ suc ) = (𝑝 ↾ suc ))
3938eqeq1d 2737 . . . . . . . . . . . . . 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 632 . . . . . . . . . . 11 (𝑢 = 𝑝 → (( ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc ) = (𝑣 ↾ suc ))) ↔ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc )))))
4342cbvrexvw 3236 . . . . . . . . . 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 3681 . . . . . . . 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 771 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → 𝑝𝐴)
51 simprrl 781 . . . . . . . . 9 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ dom 𝑝)
52 simprrr 782 . . . . . . . . 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 27767 . . . . . . . . 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 3996 . . . . . . . . . . . . . 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 27736 . . . . . . . . . . . . . . 15 <s Or No
59 soasym 5629 . . . . . . . . . . . . . . 15 (( <s Or No ∧ (𝑝 No 𝑈 No )) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6058, 59mpan 690 . . . . . . . . . . . . . 14 ((𝑝 No 𝑈 No ) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝))
6156, 57, 60syl2anc 584 . . . . . . . . . . . . 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 5994 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑈 → (𝑣 ↾ suc ) = (𝑈 ↾ suc ))
6665eqeq2d 2746 . . . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . . . 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 27710 . . . . . . . . . . . . . . . . 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 6411 . . . . . . . . . . . . . . . 16 ((dom 𝑝 ∈ On ∧ ∈ dom 𝑝) → ∈ On)
7573, 51, 74syl2anc 584 . . . . . . . . . . . . . . 15 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → ∈ On)
76 onsucb 7837 . . . . . . . . . . . . . . 15 ( ∈ On ↔ suc ∈ On)
7775, 76sylib 218 . . . . . . . . . . . . . 14 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀𝑣𝐴𝑣 <s 𝑝 → (𝑝 ↾ suc ) = (𝑣 ↾ suc ))))) → suc ∈ On)
78 noreson 27720 . . . . . . . . . . . . . 14 ((𝑈 No ∧ suc ∈ On) → (𝑈 ↾ suc ) ∈ No )
7957, 77, 78syl2anc 584 . . . . . . . . . . . . 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 5621 . . . . . . . . . . . . . 14 (( <s Or No ∧ (𝑈 ↾ suc ) ∈ No ) → ¬ (𝑈 ↾ suc ) <s (𝑈 ↾ suc ))
8158, 80mpan 690 . . . . . . . . . . . . 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 27722 . . . . . . . . . . 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 3154 . . . . . 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 240 . . . . 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 27713 . . . . . . . 8 (𝑆 No → Ord dom 𝑆)
95 ordsucss 7838 . . . . . . . 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 6028 . . . . 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 3144 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ∀ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ))
102 noresle 27757 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cun 3961  cin 3962  wss 3963  ifcif 4531  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231   Or wor 5596  dom cdm 5689  cres 5691  Ord word 6385  Oncon0 6386  suc csuc 6388  cio 6514  cfv 6563  crio 7387  2oc2o 8499   No csur 27699   <s cslt 27700
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-int 4952  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:  nosupbnd1lem2  27769  nosupbnd1lem6  27773
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