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Theorem findcard2d 9117
Description: Deduction version of findcard2 9115. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
findcard2d.ch (𝑥 = ∅ → (𝜓𝜒))
findcard2d.th (𝑥 = 𝑦 → (𝜓𝜃))
findcard2d.ta (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))
findcard2d.et (𝑥 = 𝐴 → (𝜓𝜂))
findcard2d.z (𝜑𝜒)
findcard2d.i ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))
findcard2d.a (𝜑𝐴 ∈ Fin)
Assertion
Ref Expression
findcard2d (𝜑𝜂)
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
StepHypRef Expression
1 ssid 3971 . 2 𝐴𝐴
2 findcard2d.a . . . 4 (𝜑𝐴 ∈ Fin)
32adantr 482 . . 3 ((𝜑𝐴𝐴) → 𝐴 ∈ Fin)
4 sseq1 3974 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
54anbi2d 630 . . . . 5 (𝑥 = ∅ → ((𝜑𝑥𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
6 findcard2d.ch . . . . 5 (𝑥 = ∅ → (𝜓𝜒))
75, 6imbi12d 345 . . . 4 (𝑥 = ∅ → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)))
8 sseq1 3974 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 630 . . . . 5 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 findcard2d.th . . . . 5 (𝑥 = 𝑦 → (𝜓𝜃))
119, 10imbi12d 345 . . . 4 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑𝑦𝐴) → 𝜃)))
12 sseq1 3974 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
1312anbi2d 630 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑𝑥𝐴) ↔ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)))
14 findcard2d.ta . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))
1513, 14imbi12d 345 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
16 sseq1 3974 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
1716anbi2d 630 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝐴) ↔ (𝜑𝐴𝐴)))
18 findcard2d.et . . . . 5 (𝑥 = 𝐴 → (𝜓𝜂))
1917, 18imbi12d 345 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝐴) → 𝜓) ↔ ((𝜑𝐴𝐴) → 𝜂)))
20 findcard2d.z . . . . 5 (𝜑𝜒)
2120adantr 482 . . . 4 ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)
22 simprl 770 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝜑)
23 simprr 772 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
2423unssad 4152 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦𝐴)
2522, 24jca 513 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜑𝑦𝐴))
26 id 22 . . . . . . . . . . 11 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
27 vsnid 4628 . . . . . . . . . . . 12 𝑧 ∈ {𝑧}
28 elun2 4142 . . . . . . . . . . . 12 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
2927, 28mp1i 13 . . . . . . . . . . 11 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑧 ∈ (𝑦 ∪ {𝑧}))
3026, 29sseldd 3950 . . . . . . . . . 10 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑧𝐴)
3130ad2antll 728 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧𝐴)
32 simplr 768 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧𝑦)
3331, 32eldifd 3926 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴𝑦))
34 findcard2d.i . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))
3522, 24, 33, 34syl12anc 836 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜃𝜏))
3625, 35embantd 59 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (((𝜑𝑦𝐴) → 𝜃) → 𝜏))
3736ex 414 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝜑𝑦𝐴) → 𝜃) → 𝜏)))
3837com23 86 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝜑𝑦𝐴) → 𝜃) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
397, 11, 15, 19, 21, 38findcard2s 9116 . . 3 (𝐴 ∈ Fin → ((𝜑𝐴𝐴) → 𝜂))
403, 39mpcom 38 . 2 ((𝜑𝐴𝐴) → 𝜂)
411, 40mpan2 690 1 (𝜑𝜂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cdif 3912  cun 3913  wss 3915  c0 4287  {csn 4591  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  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-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-en 8891  df-fin 8894
This theorem is referenced by:  fprodmodd  15887  sumeven  16276  sumodd  16277  maducoeval2  22005  madugsum  22008  esum2dlem  32731  fiunelcarsg  32956  carsgclctunlem1  32957  fiiuncl  43347  mpct  43496  fprodexp  43909  fprodabs2  43910  mccl  43913  fprodcn  43915  fprodcncf  44215  dvnprodlem3  44263  sge0iunmptlemfi  44728  hoidmvle  44915
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