Theorem 0elold 28065

Description: Zero is in the old set of any nonzero number. (Contributed by Scott Fenton, 13-Mar-2025.)

Hypotheses

Ref	Expression
0elold.1	⊢ (𝜑 → 𝐴 ∈ No )
0elold.2	⊢ (𝜑 → 𝐴 ≠ 0s )

Assertion

Ref	Expression
0elold	⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))

Proof of Theorem 0elold

Step	Hyp	Ref	Expression
1		bday0 27966	. . 3 ⊢ ( bday ‘ 0s ) = ∅
2		0elold.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s )
3	2	neneqd 2969	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s )
4		0elold.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
5		bday0b 27968	. . . . . 6 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s ))
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s ))
7	3, 6	mtbird 328	. . . 4 ⊢ (𝜑 → ¬ ( bday ‘𝐴) = ∅)
8		bdayon 27907	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9		on0eqel 6484	. . . . 5 ⊢ (( bday ‘𝐴) ∈ On → (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)))
10	8, 9	ax-mp 5	. . . 4 ⊢ (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴))
11		orel1 901	. . . 4 ⊢ (¬ ( bday ‘𝐴) = ∅ → ((( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)) → ∅ ∈ ( bday ‘𝐴)))
12	7, 10, 11	mpisyl 22	. . 3 ⊢ (𝜑 → ∅ ∈ ( bday ‘𝐴))
13	1, 12	eqeltrid 2873	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
14		0no 27964	. . 3 ⊢ 0s ∈ No
15		oldbday 28056	. . 3 ⊢ ((( bday ‘𝐴) ∈ On ∧ 0s ∈ No ) → ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴)))
16	8, 14, 15	mp2an 704	. 2 ⊢ ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
17	13, 16	sylibr 237	1 ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∅c0 4294  Oncon0 6358  ‘cfv 6534   No csur 27766   bday cbday 27768   0_s c0s 27960   O cold 27978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-no 27769  df-lts 27770  df-bday 27771  df-slts 27913  df-cuts 27915  df-0s 27962  df-made 27982  df-old 27983  df-left 27985  df-right 27986

This theorem is referenced by:  0elleft  28066  0elright  28067