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Theorem scutbdaylt 27179
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 1200 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s {𝑋})
2 simp2r 1201 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} <<s 𝐵)
3 snnzg 4736 . . . . . 6 (𝑋 No → {𝑋} ≠ ∅)
433ad2ant1 1134 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} ≠ ∅)
5 sslttr 27168 . . . . 5 ((𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵 ∧ {𝑋} ≠ ∅) → 𝐴 <<s 𝐵)
61, 2, 4, 5syl3anc 1372 . . . 4 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
7 scutbday 27165 . . . 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 27135 . . . . 5 bday Fn No
10 ssrab2 4038 . . . . 5 {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
11 simp1 1137 . . . . . 6 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 No )
12 simp2 1138 . . . . . 6 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵))
13 sneq 4597 . . . . . . . . 9 (𝑦 = 𝑋 → {𝑦} = {𝑋})
1413breq2d 5118 . . . . . . . 8 (𝑦 = 𝑋 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑋}))
1513breq1d 5116 . . . . . . . 8 (𝑦 = 𝑋 → ({𝑦} <<s 𝐵 ↔ {𝑋} <<s 𝐵))
1614, 15anbi12d 632 . . . . . . 7 (𝑦 = 𝑋 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
1716elrab 3646 . . . . . 6 (𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
1811, 12, 17sylanbrc 584 . . . . 5 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
19 fnfvima 7184 . . . . 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 1466 . . . 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 4925 . . . 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 3983 . 2 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑋))
24 simprl 770 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s {𝑋})
25 simprr 772 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} <<s 𝐵)
263adantr 482 . . . . . . . . . . . 12 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} ≠ ∅)
2724, 25, 26, 5syl3anc 1372 . . . . . . . . . . 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 2735 . . . . . . . . 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 2740 . . . . . . . . 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 287 . . . . . . . 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 478 . . . . . . 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 227 . . . . . . . 8 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝑋 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
3427adantr 482 . . . . . . . . 9 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → 𝐴 <<s 𝐵)
35 conway 27160 . . . . . . . . 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 6852 . . . . . . . . . 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 7340 . . . . . . . . 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 2740 . . . . . . . . 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 287 . . . . . . . 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 684 . . . . . . 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 231 . . . . . 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 27161 . . . . . . 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 2776 . . . . 5 (((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋)) → 𝑋 = (𝐴 |s 𝐵))
4645ex 414 . . . 4 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday 𝑋) → 𝑋 = (𝐴 |s 𝐵)))
4746necon3d 2961 . . 3 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (𝑋 ≠ (𝐴 |s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋)))
48473impia 1118 . 2 ((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋))
49 bdayelon 27138 . . 3 ( bday ‘(𝐴 |s 𝐵)) ∈ On
50 bdayelon 27138 . . 3 ( bday 𝑋) ∈ On
51 onelpss 6358 . . 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 691 . 2 (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday 𝑋)))
5323, 48, 52sylanbrc 584 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  ∃!wreu 3350  {crab 3406  wss 3911  c0 4283  {csn 4587   cint 4908   class class class wbr 5106  cima 5637  Oncon0 6318   Fn wfn 6492  cfv 6497  crio 7313  (class class class)co 7358   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145
This theorem is referenced by:  slerec  27180
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