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Theorem bday0b 27714
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 27708 . . . 4 0s = (∅ |s ∅)
2 snelpwi 5436 . . . . . . 7 (𝑋 No → {𝑋} ∈ 𝒫 No )
3 nulsslt 27681 . . . . . . 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 27682 . . . . . . 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 4391 . . . . . . . . 9 ∅ ⊆ ( bday 𝑥)
119, 10eqsstrdi 4031 . . . . . . . 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 3144 . . . . 5 ((𝑋 No ∧ ( bday 𝑋) = ∅) → ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))
15 0elpw 5347 . . . . . . . 8 ∅ ∈ 𝒫 No
16 nulssgt 27682 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
1715, 16ax-mp 5 . . . . . . 7 ∅ <<s ∅
18 eqscut2 27690 . . . . . . 7 ((∅ <<s ∅ ∧ 𝑋 No ) → ((∅ |s ∅) = 𝑋 ↔ (∅ <<s {𝑋} ∧ {𝑋} <<s ∅ ∧ ∀𝑥 No ((∅ <<s {𝑥} ∧ {𝑥} <<s ∅) → ( bday 𝑋) ⊆ ( bday 𝑥)))))
1917, 18mpan 687 . . . . . 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 1339 . . . 4 ((𝑋 No ∧ ( bday 𝑋) = ∅) → (∅ |s ∅) = 𝑋)
221, 21eqtr2id 2779 . . 3 ((𝑋 No ∧ ( bday 𝑋) = ∅) → 𝑋 = 0s )
2322ex 412 . 2 (𝑋 No → (( bday 𝑋) = ∅ → 𝑋 = 0s ))
24 fveq2 6884 . . 3 (𝑋 = 0s → ( bday 𝑋) = ( bday ‘ 0s ))
25 bday0s 27712 . . 3 ( bday ‘ 0s ) = ∅
2624, 25eqtrdi 2782 . 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 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wss 3943  c0 4317  𝒫 cpw 4597  {csn 4623   class class class wbr 5141  cfv 6536  (class class class)co 7404   No csur 27524   bday cbday 27526   <<s csslt 27664   |s cscut 27666   0s c0s 27706
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1o 8464  df-2o 8465  df-no 27527  df-slt 27528  df-bday 27529  df-sslt 27665  df-scut 27667  df-0s 27708
This theorem is referenced by:  bday1s  27715  cuteq1  27717  0elold  27786
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