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Theorem dif1en 9198
Description: If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5370. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
Assertion
Ref Expression
dif1en ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 simp1 1135 . . 3 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → 𝐴 ≈ suc 𝑀)
2 encv 8991 . . . . 5 (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V))
32simpld 494 . . . 4 (𝐴 ≈ suc 𝑀𝐴 ∈ V)
433anim1i 1151 . . 3 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On))
5 bren 8993 . . . 4 (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀)
6 sucidg 6466 . . . . . . . . . . . . 13 (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
7 f1ocnvdm 7304 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
873adant2 1130 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
9 f1ofvswap 7325 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴 ∧ (𝑓𝑀) ∈ 𝐴) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
108, 9syld3an3 1408 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
11 f1ocnvfv2 7296 . . . . . . . . . . . . . . . . . . 19 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓‘(𝑓𝑀)) = 𝑀)
1211opeq2d 4884 . . . . . . . . . . . . . . . . . 18 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(𝑓𝑀))⟩ = ⟨𝑋, 𝑀⟩)
1312preq1d 4743 . . . . . . . . . . . . . . . . 17 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} = {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
1413uneq2d 4177 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) = ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
1514f1oeq1d 6843 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
16153adant2 1130 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
1710, 16mpbid 232 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
186, 17syl3an3 1164 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
19183adant3r1 1181 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
20 f1ofun 6850 . . . . . . . . . . 11 (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 → Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
21 opex 5474 . . . . . . . . . . . . . 14 𝑋, 𝑀⟩ ∈ V
2221prid1 4766 . . . . . . . . . . . . 13 𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}
23 elun2 4192 . . . . . . . . . . . . 13 (⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} → ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
2422, 23ax-mp 5 . . . . . . . . . . . 12 𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
25 funopfv 6958 . . . . . . . . . . . 12 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀))
2624, 25mpi 20 . . . . . . . . . . 11 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
2719, 20, 263syl 18 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
28 simpr2 1194 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → 𝑋𝐴)
29 f1ocnvfv 7297 . . . . . . . . . . 11 ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀𝑋𝐴) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
3019, 28, 29syl2anc 584 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
3127, 30mpd 15 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋)
3231sneqd 4642 . . . . . . . 8 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)} = {𝑋})
3332difeq2d 4135 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) = (𝐴 ∖ {𝑋}))
34 simpr1 1193 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → 𝐴 ∈ V)
35 3simpc 1149 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝑋𝐴𝑀 ∈ On))
3635anim2i 617 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ On)))
37 3anass 1094 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On) ↔ (𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ On)))
3836, 37sylibr 234 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On))
3934, 38jca 511 . . . . . . . 8 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)))
40 simpl 482 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → 𝐴 ∈ V)
41 simpr3 1195 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → 𝑀 ∈ On)
4240, 41jca 511 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝐴 ∈ V ∧ 𝑀 ∈ On))
43 simpr 484 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On))
4442, 43jca 511 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → ((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)))
45 vex 3481 . . . . . . . . . . . 12 𝑓 ∈ V
4645resex 6048 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∈ V
47 prex 5442 . . . . . . . . . . 11 {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} ∈ V
4846, 47unex 7762 . . . . . . . . . 10 ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V
49 dif1enlem 9194 . . . . . . . . . 10 (((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5048, 49mp3anl1 1454 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5118, 50sylan2 593 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ (𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5239, 44, 513syl 18 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
5333, 52eqbrtrrd 5171 . . . . . 6 ((𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On)) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
5453ex 412 . . . . 5 (𝑓:𝐴1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5554exlimiv 1927 . . . 4 (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
565, 55sylbi 217 . . 3 (𝐴 ≈ suc 𝑀 → ((𝐴 ∈ V ∧ 𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀))
571, 4, 56sylc 65 . 2 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ On) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
58573comr 1124 1 ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  cdif 3959  cun 3960  {csn 4630  {cpr 4632  cop 4636   class class class wbr 5147  ccnv 5687  cres 5690  Oncon0 6385  suc csuc 6387  Fun wfun 6556  1-1-ontowf1o 6561  cfv 6562  cen 8980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-en 8984
This theorem is referenced by:  dif1ennn  9199
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