Theorem 0elold 27821

Description: Zero is in the old set of any non-zero 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		bday0s 27740	. . 3 ⊢ ( bday ‘ 0s ) = ∅
2		0elold.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s )
3	2	neneqd 2930	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s )
4		0elold.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
5		bday0b 27742	. . . . . 6 ⊢ (𝐴 ∈ No → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s ))
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → (( bday ‘𝐴) = ∅ ↔ 𝐴 = 0s ))
7	3, 6	mtbird 325	. . . 4 ⊢ (𝜑 → ¬ ( bday ‘𝐴) = ∅)
8		bdayelon 27688	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9		on0eqel 6458	. . . . 5 ⊢ (( bday ‘𝐴) ∈ On → (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)))
10	8, 9	ax-mp 5	. . . 4 ⊢ (( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴))
11		orel1 888	. . . 4 ⊢ (¬ ( bday ‘𝐴) = ∅ → ((( bday ‘𝐴) = ∅ ∨ ∅ ∈ ( bday ‘𝐴)) → ∅ ∈ ( bday ‘𝐴)))
12	7, 10, 11	mpisyl 21	. . 3 ⊢ (𝜑 → ∅ ∈ ( bday ‘𝐴))
13	1, 12	eqeltrid 2832	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
14		0sno 27738	. . 3 ⊢ 0s ∈ No
15		oldbday 27812	. . 3 ⊢ ((( bday ‘𝐴) ∈ On ∧ 0s ∈ No ) → ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴)))
16	8, 14, 15	mp2an 692	. 2 ⊢ ( 0s ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
17	13, 16	sylibr 234	1 ⊢ (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  Oncon0 6332  ‘cfv 6511   No csur 27551   bday cbday 27553   0_s c0s 27734   O cold 27751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sslt 27693  df-scut 27695  df-0s 27736  df-made 27755  df-old 27756  df-left 27758  df-right 27759

This theorem is referenced by:  0elleft  27822  0elright  27823