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Theorem bday0b 27875
Description: The only surreal with birthday is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
bday0b (𝑋 No → (( bday 𝑋) = ∅ ↔ 𝑋 = 0s ))

Proof of Theorem bday0b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-0s 27869 . . . 4 0s = (∅ |s ∅)
2 snelpwi 5448 . . . . . . 7 (𝑋 No → {𝑋} ∈ 𝒫 No )
3 nulsslt 27842 . . . . . . 7 ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋})
42, 3syl 17 . . . . . 6 (𝑋 No → ∅ <<s {𝑋})
54adantr 480 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∅ <<s {𝑋})
6 nulssgt 27843 . . . . . . 7 ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅)
72, 6syl 17 . . . . . 6 (𝑋 No → {𝑋} <<s ∅)
87adantr 480 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → {𝑋} <<s ∅)
9 id 22 . . . . . . . . 9 (( bday 𝑋) = ∅ → ( bday 𝑋) = ∅)
10 0ss 4400 . . . . . . . . 9 ∅ ⊆ ( bday 𝑥)
119, 10eqsstrdi 4028 . . . . . . . 8 (( bday 𝑋) = ∅ → ( bday 𝑋) ⊆ ( bday 𝑥))
1211a1d 25 . . . . . . 7 (( bday 𝑋) = ∅ → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1312adantl 481 . . . . . 6 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1413ralrimivw 3150 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
15 0elpw 5356 . . . . . . . 8 ∅ ∈ 𝒫 No
16 nulssgt 27843 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
1715, 16ax-mp 5 . . . . . . 7 ∅ <<s ∅
18 eqscut2 27851 . . . . . . 7 ((∅ <<s ∅ ∧ 𝑋 No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
1917, 18mpan 690 . . . . . 6 (𝑋 No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
2019adantr 480 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
215, 8, 14, 20mpbir3and 1343 . . . 4 ((𝑋 No ∧ ( bday 𝑋) = ∅) → (∅ |s ∅) = 𝑋)
221, 21eqtr2id 2790 . . 3 ((𝑋 No ∧ ( bday 𝑋) = ∅) → 𝑋 = 0s )
2322ex 412 . 2 (𝑋 No → (( bday 𝑋) = ∅ → 𝑋 = 0s ))
24 fveq2 6906 . . 3 (𝑋 = 0s → ( bday 𝑋) = ( bday ‘ 0s ))
25 bday0s 27873 . . 3 ( bday ‘ 0s ) = ∅
2624, 25eqtrdi 2793 . 2 (𝑋 = 0s → ( bday 𝑋) = ∅)
2723, 26impbid1 225 1 (𝑋 No → (( bday 𝑋) = ∅ ↔ 𝑋 = 0s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   bday cbday 27686   <<s csslt 27825   |s cscut 27827   0s c0s 27867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869
This theorem is referenced by:  bday1s  27876  cuteq1  27878  0elold  27947
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