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Theorem scutbdaylt 33783
Description: If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)
Assertion
Ref Expression
scutbdaylt ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋))

Proof of Theorem scutbdaylt
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2l 1201 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s {𝑋})
2 simp2r 1202 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} <<s 𝐵)
3 snnzg 4707 . . . . . 6 (𝑋 No → {𝑋} ≠ ∅)
433ad2ant1 1135 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} ≠ ∅)
5 sslttr 33772 . . . . 5 ((𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵 ∧ {𝑋} ≠ ∅) → 𝐴 <<s 𝐵)
61, 2, 4, 5syl3anc 1373 . . . 4 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
7 scutbday 33769 . . . 4 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
86, 7syl 17 . . 3 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
9 bdayfn 33739 . . . . 5 bday Fn No
10 ssrab2 4010 . . . . 5 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
11 simp1 1138 . . . . . 6 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 No )
12 simp2 1139 . . . . . 6 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵))
13 sneq 4568 . . . . . . . . 9 (𝑦 = 𝑋 → {𝑦} = {𝑋})
1413breq2d 5082 . . . . . . . 8 (𝑦 = 𝑋 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑋}))
1513breq1d 5080 . . . . . . . 8 (𝑦 = 𝑋 → ({𝑦} <<s 𝐵 ↔ {𝑋} <<s 𝐵))
1614, 15anbi12d 634 . . . . . . 7 (𝑦 = 𝑋 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
1716elrab 3617 . . . . . 6 (𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
1811, 12, 17sylanbrc 586 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
19 fnfvima 7071 . . . . 5 (( bday Fn No ∧ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No 𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday 𝑋) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
209, 10, 18, 19mp3an12i 1467 . . . 4 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday 𝑋) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
21 intss1 4891 . . . 4 (( bday 𝑋) ∈ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑋))
2220, 21syl 17 . . 3 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday 𝑋))
238, 22eqsstrd 3956 . 2 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑋))
24 simprl 771 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s {𝑋})
25 simprr 773 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} <<s 𝐵)
263adantr 484 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} ≠ ∅)
2724, 25, 26, 5syl3anc 1373 . . . . . . . . . . 11 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s 𝐵)
2827, 7syl 17 . . . . . . . . . 10 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
2928eqeq1d 2741 . . . . . . . . 9 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋) ↔ ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) = ( bday 𝑋)))
30 eqcom 2746 . . . . . . . . 9 ( ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) = ( bday 𝑋) ↔ ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3129, 30bitrdi 290 . . . . . . . 8 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋) ↔ ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
3231biimpa 480 . . . . . . 7 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3317biimpri 231 . . . . . . . 8 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
3427adantr 484 . . . . . . . . 9 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → 𝐴 <<s 𝐵)
35 conway 33764 . . . . . . . . 9 (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3634, 35syl 17 . . . . . . . 8 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
37 fveqeq2 6748 . . . . . . . . . 10 (𝑥 = 𝑋 → (( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
3837riota2 7218 . . . . . . . . 9 ((𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ∧ ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) → (( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) = 𝑋))
39 eqcom 2746 . . . . . . . . 9 ((𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) = 𝑋𝑋 = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
4038, 39bitrdi 290 . . . . . . . 8 ((𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ∧ ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) → (( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ 𝑋 = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))))
4133, 36, 40syl2an2r 685 . . . . . . 7 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → (( bday 𝑋) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ 𝑋 = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))))
4232, 41mpbid 235 . . . . . 6 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → 𝑋 = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
43 scutval 33765 . . . . . . 7 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
4434, 43syl 17 . . . . . 6 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → (𝐴 |s 𝐵) = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
4542, 44eqtr4d 2782 . . . . 5 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → 𝑋 = (𝐴 |s 𝐵))
4645ex 416 . . . 4 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋) → 𝑋 = (𝐴 |s 𝐵)))
4746necon3d 2964 . . 3 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (𝑋 ≠ (𝐴 |s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋)))
48473impia 1119 . 2 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋))
49 bdayelon 33742 . . 3 ( bday ‘(𝐴 |s 𝐵)) ∈ On
50 bdayelon 33742 . . 3 ( bday 𝑋) ∈ On
51 onelpss 6274 . . 3 ((( bday ‘(𝐴 |s 𝐵)) ∈ On ∧ ( bday 𝑋) ∈ On) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋))))
5249, 50, 51mp2an 692 . 2 (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋)))
5323, 48, 52sylanbrc 586 1 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2112  wne 2943  ∃!wreu 3066  {crab 3068  wss 3883  c0 4254  {csn 4558   cint 4876   class class class wbr 5070  cima 5572  Oncon0 6234   Fn wfn 6396  cfv 6401  crio 7191  (class class class)co 7235   No csur 33614   bday cbday 33616   <<s csslt 33746   |s cscut 33748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5196  ax-sep 5209  ax-nul 5216  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5179  df-id 5472  df-eprel 5478  df-po 5486  df-so 5487  df-fr 5527  df-we 5529  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-ord 6237  df-on 6238  df-suc 6240  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-f1 6406  df-fo 6407  df-f1o 6408  df-fv 6409  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-1o 8226  df-2o 8227  df-no 33617  df-slt 33618  df-bday 33619  df-sslt 33747  df-scut 33749
This theorem is referenced by:  slerec  33784
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