Theorem 0elold 27878

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 27797	. . 3 ⊢ ( bday ‘ 0s ) = ∅
2		0elold.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s )
3	2	neneqd 2938	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s )
4		0elold.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
5		bday0b 27799	. . . . . 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 27745	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9		on0eqel 6483	. . . . 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 2839	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
14		0sno 27795	. . 3 ⊢ 0s ∈ No
15		oldbday 27869	. . 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 2933  ∅c0 4313  Oncon0 6357  ‘cfv 6536   No csur 27608   bday cbday 27610   0_s c0s 27791   O cold 27808

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  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 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sslt 27750  df-scut 27752  df-0s 27793  df-made 27812  df-old 27813  df-left 27815  df-right 27816

This theorem is referenced by:  0elleft  27879  0elright  27880