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Theorem 0elold 27948
Description: Zero is in the old set of any non-zero number. (Contributed by Scott Fenton, 13-Mar-2025.)
Hypotheses
Ref Expression
0elold.1 (𝜑𝐴 No )
0elold.2 (𝜑𝐴 ≠ 0s )
Assertion
Ref Expression
0elold (𝜑 → 0s ∈ ( O ‘( bday 𝐴)))

Proof of Theorem 0elold
StepHypRef Expression
1 bday0s 27874 . . 3 ( bday ‘ 0s ) = ∅
2 0elold.2 . . . . . 6 (𝜑𝐴 ≠ 0s )
32neneqd 2944 . . . . 5 (𝜑 → ¬ 𝐴 = 0s )
4 0elold.1 . . . . . 6 (𝜑𝐴 No )
5 bday0b 27876 . . . . . 6 (𝐴 No → (( bday 𝐴) = ∅ ↔ 𝐴 = 0s ))
64, 5syl 17 . . . . 5 (𝜑 → (( bday 𝐴) = ∅ ↔ 𝐴 = 0s ))
73, 6mtbird 325 . . . 4 (𝜑 → ¬ ( bday 𝐴) = ∅)
8 bdayelon 27822 . . . . 5 ( bday 𝐴) ∈ On
9 on0eqel 6507 . . . . 5 (( bday 𝐴) ∈ On → (( bday 𝐴) = ∅ ∨ ∅ ∈ ( bday 𝐴)))
108, 9ax-mp 5 . . . 4 (( bday 𝐴) = ∅ ∨ ∅ ∈ ( bday 𝐴))
11 orel1 888 . . . 4 (¬ ( bday 𝐴) = ∅ → ((( bday 𝐴) = ∅ ∨ ∅ ∈ ( bday 𝐴)) → ∅ ∈ ( bday 𝐴)))
127, 10, 11mpisyl 21 . . 3 (𝜑 → ∅ ∈ ( bday 𝐴))
131, 12eqeltrid 2844 . 2 (𝜑 → ( bday ‘ 0s ) ∈ ( bday 𝐴))
14 0sno 27872 . . 3 0s No
15 oldbday 27940 . . 3 ((( bday 𝐴) ∈ On ∧ 0s No ) → ( 0s ∈ ( O ‘( bday 𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday 𝐴)))
168, 14, 15mp2an 692 . 2 ( 0s ∈ ( O ‘( bday 𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday 𝐴))
1713, 16sylibr 234 1 (𝜑 → 0s ∈ ( O ‘( bday 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 847   = wceq 1539  wcel 2107  wne 2939  c0 4332  Oncon0 6383  cfv 6560   No csur 27685   bday cbday 27687   0s c0s 27868   O cold 27883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891
This theorem is referenced by:  0elleft  27949  0elright  27950
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