Theorem 0elold 28353

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 28254	. . 3 ⊢ (bday‘ 0s ) = ∅
2		0elold.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s )
3	2	neneqd 2964	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s )
4		0elold.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No)
5		bday0b 28256	. . . . . 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 28195	. . . . 5 ⊢ (bday‘𝐴) ∈ On
9		on0eqel 6514	. . . . 5 ⊢ ((bday‘𝐴) ∈ On → ((bday‘𝐴) = ∅ ∨ ∅ ∈ (bday‘𝐴)))
10	8, 9	ax-mp 5	. . . 4 ⊢ ((bday‘𝐴) = ∅ ∨ ∅ ∈ (bday‘𝐴))
11		orel1 902	. . . 4 ⊢ (¬ (bday‘𝐴) = ∅ → (((bday‘𝐴) = ∅ ∨ ∅ ∈ (bday‘𝐴)) → ∅ ∈ (bday‘𝐴)))
12	7, 10, 11	mpisyl 22	. . 3 ⊢ (𝜑 → ∅ ∈ (bday‘𝐴))
13	1, 12	eqeltrid 2868	. 2 ⊢ (𝜑 → (bday‘ 0s ) ∈ (bday‘𝐴))
14		0no 28252	. . 3 ⊢ 0s ∈ No
15		oldbday 28344	. . 3 ⊢ (((bday‘𝐴) ∈ On ∧ 0s ∈ No) → ( 0s ∈ (O‘(bday‘𝐴)) ↔ (bday‘ 0s ) ∈ (bday‘𝐴)))
16	8, 14, 15	mp2an 705	. 2 ⊢ ( 0s ∈ (O‘(bday‘𝐴)) ↔ (bday‘ 0s ) ∈ (bday‘𝐴))
17	13, 16	sylibr 237	1 ⊢ (𝜑 → 0s ∈ (O‘(bday‘𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  Oncon0 6388  ‘cfv 6566  Nocsur 28054  bdaycbday 28056   0_s c0s 28248  Ocold 28266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274

This theorem is used by:  0elleft  28354  0elright  28355