Theorem findcard2d 9165

Description: Deduction version of findcard2 9163. (Contributed by SO, 16-Jul-2018.)

Hypotheses

Ref	Expression
findcard2d.ch	⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
findcard2d.th	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
findcard2d.ta	⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
findcard2d.et	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
findcard2d.z	⊢ (𝜑 → 𝜒)
findcard2d.i	⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
findcard2d.a	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
findcard2d	⊢ (𝜑 → 𝜂)

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜂,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦, 𝑧)   𝜃(𝑦, 𝑧)   𝜏(𝑦, 𝑧)   𝜂(𝑦, 𝑧)

Proof of Theorem findcard2d

Step	Hyp	Ref	Expression
1		ssid 3956	. 2 ⊢ 𝐴 ⊆ 𝐴
2		findcard2d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
3	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝐴 ∈ Fin)
4		sseq1 3959	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
5	4	anbi2d 642	. . . . 5 ⊢ (𝑥 = ∅ → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
6		findcard2d.ch	. . . . 5 ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑥 = ∅ → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)))
8		sseq1 3959	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9	8	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑦 ⊆ 𝐴)))
10		findcard2d.th	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
11	9, 10	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃)))
12		sseq1 3959	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
13	12	anbi2d 642	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)))
14		findcard2d.ta	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
15	13, 14	imbi12d 347	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
16		sseq1 3959	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
17	16	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
18		findcard2d.et	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
19	17, 18	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)))
20		findcard2d.z	. . . . 5 ⊢ (𝜑 → 𝜒)
21	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)
22		simprl 783	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝜑)
23		simprr 785	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
24	23	unssad 4142	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦 ⊆ 𝐴)
25	22, 24	jca 521	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜑 ∧ 𝑦 ⊆ 𝐴))
26		id 23	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
27		vsnid 4627	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ {𝑧}
28		elun2 4132	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29	27, 28	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ (𝑦 ∪ {𝑧}))
30	26, 29	sseldd 3935	. . . . . . . . . 10 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ 𝐴)
31	30	ad2antll 742	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
32		simplr 781	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑦)
33	31, 32	eldifd 3913	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴 ∖ 𝑦))
34		findcard2d.i	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
35	22, 24, 33, 34	syl12anc 850	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜃 → 𝜏))
36	25, 35	embantd 60	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏))
37	36	ex 418	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏)))
38	37	com23 87	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
39	7, 11, 15, 19, 21, 38	findcard2s 9164	. . 3 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂))
40	3, 39	mpcom 39	. 2 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)
41	1, 40	mpan2 704	1 ⊢ (𝜑 → 𝜂)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587  Fincfn 8956

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  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-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7867  df-en 8957  df-fin 8960

This theorem is used by:  fprodmodd  16090  sumeven  16483  sumodd  16484  maducoeval2  22868  madugsum  22871  elrgspnlem4  33693  domnprodeq0  33727  rprmdvdsprod  33952  deg1prod  34001  mplidomlem  34045  psrgsum  34066  psrmonprod  34070  vieta  34098  constrextdg2lem  34266  constrfiss  34269  esum2dlem  34610  fiunelcarsg  34835  carsgclctunlem1  34836  evl1gprodd  42991  idomnnzgmulnz  43007  deg1gprod  43014  fiiuncl  45907  mpct  46040  fprodexp  46432  fprodabs2  46433  mccl  46436  fprodcn  46438  fprodcncf  46736  dvnprodlem3  46784  sge0iunmptlemfi  47249  hoidmvle  47436