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Theorem cuteq1 27893
Description: Condition for a surreal cut to equal one. (Contributed by Scott Fenton, 12-Mar-2025.)
Hypotheses
Ref Expression
cuteq1.1 (𝜑 → 0s𝐴)
cuteq1.2 (𝜑𝐴 <<s { 1s })
cuteq1.3 (𝜑 → { 1s } <<s 𝐵)
Assertion
Ref Expression
cuteq1 (𝜑 → (𝐴 |s 𝐵) = 1s )

Proof of Theorem cuteq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cuteq1.2 . 2 (𝜑𝐴 <<s { 1s })
2 cuteq1.3 . 2 (𝜑 → { 1s } <<s 𝐵)
3 bday1s 27891 . . . . . 6 ( bday ‘ 1s ) = 1o
4 df-1o 8505 . . . . . 6 1o = suc ∅
53, 4eqtri 2763 . . . . 5 ( bday ‘ 1s ) = suc ∅
6 ssltsep 27850 . . . . . . . . . . . . . 14 (𝐴 <<s { 0s } → ∀𝑥𝐴𝑦 ∈ { 0s }𝑥 <s 𝑦)
7 dfral2 3097 . . . . . . . . . . . . . . . 16 (∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
87ralbii 3091 . . . . . . . . . . . . . . 15 (∀𝑥𝐴𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ∀𝑥𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
9 ralnex 3070 . . . . . . . . . . . . . . 15 (∀𝑥𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ ∃𝑥𝐴𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
108, 9bitri 275 . . . . . . . . . . . . . 14 (∀𝑥𝐴𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑥𝐴𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
116, 10sylib 218 . . . . . . . . . . . . 13 (𝐴 <<s { 0s } → ¬ ∃𝑥𝐴𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
12 cuteq1.1 . . . . . . . . . . . . . . 15 (𝜑 → 0s𝐴)
13 0sno 27886 . . . . . . . . . . . . . . . 16 0s No
14 sltirr 27806 . . . . . . . . . . . . . . . 16 ( 0s No → ¬ 0s <s 0s )
1513, 14ax-mp 5 . . . . . . . . . . . . . . 15 ¬ 0s <s 0s
16 breq1 5151 . . . . . . . . . . . . . . . . 17 (𝑥 = 0s → (𝑥 <s 0s ↔ 0s <s 0s ))
1716notbid 318 . . . . . . . . . . . . . . . 16 (𝑥 = 0s → (¬ 𝑥 <s 0s ↔ ¬ 0s <s 0s ))
1817rspcev 3622 . . . . . . . . . . . . . . 15 (( 0s𝐴 ∧ ¬ 0s <s 0s ) → ∃𝑥𝐴 ¬ 𝑥 <s 0s )
1912, 15, 18sylancl 586 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑥𝐴 ¬ 𝑥 <s 0s )
2013elexi 3501 . . . . . . . . . . . . . . . 16 0s ∈ V
21 breq2 5152 . . . . . . . . . . . . . . . . 17 (𝑦 = 0s → (𝑥 <s 𝑦𝑥 <s 0s ))
2221notbid 318 . . . . . . . . . . . . . . . 16 (𝑦 = 0s → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s ))
2320, 22rexsn 4687 . . . . . . . . . . . . . . 15 (∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s )
2423rexbii 3092 . . . . . . . . . . . . . 14 (∃𝑥𝐴𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ∃𝑥𝐴 ¬ 𝑥 <s 0s )
2519, 24sylibr 234 . . . . . . . . . . . . 13 (𝜑 → ∃𝑥𝐴𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
2611, 25nsyl3 138 . . . . . . . . . . . 12 (𝜑 → ¬ 𝐴 <<s { 0s })
2726adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 No ) → ¬ 𝐴 <<s { 0s })
28 sneq 4641 . . . . . . . . . . . . 13 (𝑥 = 0s → {𝑥} = { 0s })
2928breq2d 5160 . . . . . . . . . . . 12 (𝑥 = 0s → (𝐴 <<s {𝑥} ↔ 𝐴 <<s { 0s }))
3029notbid 318 . . . . . . . . . . 11 (𝑥 = 0s → (¬ 𝐴 <<s {𝑥} ↔ ¬ 𝐴 <<s { 0s }))
3127, 30syl5ibrcom 247 . . . . . . . . . 10 ((𝜑𝑥 No ) → (𝑥 = 0s → ¬ 𝐴 <<s {𝑥}))
3231necon2ad 2953 . . . . . . . . 9 ((𝜑𝑥 No ) → (𝐴 <<s {𝑥} → 𝑥 ≠ 0s ))
3332adantrd 491 . . . . . . . 8 ((𝜑𝑥 No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → 𝑥 ≠ 0s ))
3433impr 454 . . . . . . 7 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → 𝑥 ≠ 0s )
35 bday0b 27890 . . . . . . . . 9 (𝑥 No → (( bday 𝑥) = ∅ ↔ 𝑥 = 0s ))
3635ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday 𝑥) = ∅ ↔ 𝑥 = 0s ))
3736necon3bid 2983 . . . . . . 7 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday 𝑥) ≠ ∅ ↔ 𝑥 ≠ 0s ))
3834, 37mpbird 257 . . . . . 6 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday 𝑥) ≠ ∅)
39 bdayelon 27836 . . . . . . . . 9 ( bday 𝑥) ∈ On
4039onordi 6497 . . . . . . . 8 Ord ( bday 𝑥)
41 ord0eln0 6441 . . . . . . . 8 (Ord ( bday 𝑥) → (∅ ∈ ( bday 𝑥) ↔ ( bday 𝑥) ≠ ∅))
4240, 41ax-mp 5 . . . . . . 7 (∅ ∈ ( bday 𝑥) ↔ ( bday 𝑥) ≠ ∅)
43 0elon 6440 . . . . . . . 8 ∅ ∈ On
4443, 39onsucssi 7862 . . . . . . 7 (∅ ∈ ( bday 𝑥) ↔ suc ∅ ⊆ ( bday 𝑥))
4542, 44bitr3i 277 . . . . . 6 (( bday 𝑥) ≠ ∅ ↔ suc ∅ ⊆ ( bday 𝑥))
4638, 45sylib 218 . . . . 5 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → suc ∅ ⊆ ( bday 𝑥))
475, 46eqsstrid 4044 . . . 4 ((𝜑 ∧ (𝑥 No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘ 1s ) ⊆ ( bday 𝑥))
4847expr 456 . . 3 ((𝜑𝑥 No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday 𝑥)))
4948ralrimiva 3144 . 2 (𝜑 → ∀𝑥 No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday 𝑥)))
50 1sno 27887 . . . . . . 7 1s No
5150elexi 3501 . . . . . 6 1s ∈ V
5251snnz 4781 . . . . 5 { 1s } ≠ ∅
53 sslttr 27867 . . . . 5 ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ { 1s } ≠ ∅) → 𝐴 <<s 𝐵)
5452, 53mp3an3 1449 . . . 4 ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵) → 𝐴 <<s 𝐵)
551, 2, 54syl2anc 584 . . 3 (𝜑𝐴 <<s 𝐵)
56 eqscut2 27866 . . 3 ((𝐴 <<s 𝐵 ∧ 1s No ) → ((𝐴 |s 𝐵) = 1s ↔ (𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ ∀𝑥 No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday 𝑥)))))
5755, 50, 56sylancl 586 . 2 (𝜑 → ((𝐴 |s 𝐵) = 1s ↔ (𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ ∀𝑥 No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday 𝑥)))))
581, 2, 49, 57mpbir3and 1341 1 (𝜑 → (𝐴 |s 𝐵) = 1s )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  Ord word 6385  suc csuc 6388  cfv 6563  (class class class)co 7431  1oc1o 8498   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   |s cscut 27842   0s c0s 27882   1s c1s 27883
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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885
This theorem is referenced by:  precsexlem11  28256
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