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Theorem 0elold 27962
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 27888 . . 3 ( bday ‘ 0s ) = ∅
2 0elold.2 . . . . . 6 (𝜑𝐴 ≠ 0s )
32neneqd 2943 . . . . 5 (𝜑 → ¬ 𝐴 = 0s )
4 0elold.1 . . . . . 6 (𝜑𝐴 No )
5 bday0b 27890 . . . . . 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 27836 . . . . 5 ( bday 𝐴) ∈ On
9 on0eqel 6510 . . . . 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 2843 . 2 (𝜑 → ( bday ‘ 0s ) ∈ ( bday 𝐴))
14 0sno 27886 . . 3 0s No
15 oldbday 27954 . . 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 1537  wcel 2106  wne 2938  c0 4339  Oncon0 6386  cfv 6563   No csur 27699   bday cbday 27701   0s c0s 27882   O cold 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905
This theorem is referenced by:  0elleft  27963  0elright  27964
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