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Theorem bday0b 27781
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 27775 . . . 4 0s = (∅ |s ∅)
2 snelpwi 5447 . . . . . . 7 (𝑋 No → {𝑋} ∈ 𝒫 No )
3 nulsslt 27748 . . . . . . 7 ({𝑋} ∈ 𝒫 No → ∅ <<s {𝑋})
42, 3syl 17 . . . . . 6 (𝑋 No → ∅ <<s {𝑋})
54adantr 479 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∅ <<s {𝑋})
6 nulssgt 27749 . . . . . . 7 ({𝑋} ∈ 𝒫 No → {𝑋} <<s ∅)
72, 6syl 17 . . . . . 6 (𝑋 No → {𝑋} <<s ∅)
87adantr 479 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → {𝑋} <<s ∅)
9 id 22 . . . . . . . . 9 (( bday 𝑋) = ∅ → ( bday 𝑋) = ∅)
10 0ss 4398 . . . . . . . . 9 ∅ ⊆ ( bday 𝑥)
119, 10eqsstrdi 4034 . . . . . . . 8 (( bday 𝑋) = ∅ → ( bday 𝑋) ⊆ ( bday 𝑥))
1211a1d 25 . . . . . . 7 (( bday 𝑋) = ∅ → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1312adantl 480 . . . . . 6 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
1413ralrimivw 3146 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
15 0elpw 5358 . . . . . . . 8 ∅ ∈ 𝒫 No
16 nulssgt 27749 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
1715, 16ax-mp 5 . . . . . . 7 ∅ <<s ∅
18 eqscut2 27757 . . . . . . 7 ((∅ <<s ∅ ∧ 𝑋 No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
1917, 18mpan 688 . . . . . 6 (𝑋 No → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
2019adantr 479 . . . . 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 1339 . . . 4 ((𝑋 No ∧ ( bday 𝑋) = ∅) → (∅ |s ∅) = 𝑋)
221, 21eqtr2id 2780 . . 3 ((𝑋 No ∧ ( bday 𝑋) = ∅) → 𝑋 = 0s )
2322ex 411 . 2 (𝑋 No → (( bday 𝑋) = ∅ → 𝑋 = 0s ))
24 fveq2 6900 . . 3 (𝑋 = 0s → ( bday 𝑋) = ( bday ‘ 0s ))
25 bday0s 27779 . . 3 ( bday ‘ 0s ) = ∅
2624, 25eqtrdi 2783 . 2 (𝑋 = 0s → ( bday 𝑋) = ∅)
2723, 26impbid1 224 1 (𝑋 No → (( bday 𝑋) = ∅ ↔ 𝑋 = 0s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3057  wss 3947  c0 4324  𝒫 cpw 4604  {csn 4630   class class class wbr 5150  cfv 6551  (class class class)co 7424   No csur 27591   bday cbday 27593   <<s csslt 27731   |s cscut 27733   0s c0s 27773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1o 8491  df-2o 8492  df-no 27594  df-slt 27595  df-bday 27596  df-sslt 27732  df-scut 27734  df-0s 27775
This theorem is referenced by:  bday1s  27782  cuteq1  27784  0elold  27853
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