Theorem 0elold 27828

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 27747	. . 3 ⊢ ( bday ‘ 0s ) = ∅
2		0elold.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 0s )
3	2	neneqd 2931	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 0s )
4		0elold.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
5		bday0b 27749	. . . . . 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 27695	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
9		on0eqel 6461	. . . . 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 2833	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) ∈ ( bday ‘𝐴))
14		0sno 27745	. . 3 ⊢ 0s ∈ No
15		oldbday 27819	. . 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 2926  ∅c0 4299  Oncon0 6335  ‘cfv 6514   No csur 27558   bday cbday 27560   0_s c0s 27741   O cold 27758

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766

This theorem is referenced by:  0elleft  27829  0elright  27830