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Mirrors > Home > MPE Home > Th. List > findcard2s | Structured version Visualization version GIF version |
Description: Variation of findcard2 8761 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.) |
Ref | Expression |
---|---|
findcard2s.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
findcard2s.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
findcard2s.3 | ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) |
findcard2s.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
findcard2s.5 | ⊢ 𝜓 |
findcard2s.6 | ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
findcard2s | ⊢ (𝐴 ∈ Fin → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | findcard2s.1 | . 2 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
2 | findcard2s.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
3 | findcard2s.3 | . 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) | |
4 | findcard2s.4 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
5 | findcard2s.5 | . 2 ⊢ 𝜓 | |
6 | findcard2s.6 | . . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃)) | |
7 | 6 | ex 415 | . . 3 ⊢ (𝑦 ∈ Fin → (¬ 𝑧 ∈ 𝑦 → (𝜒 → 𝜃))) |
8 | uncom 4132 | . . . . . . 7 ⊢ ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧}) | |
9 | snssi 4744 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦) | |
10 | ssequn1 4159 | . . . . . . . 8 ⊢ ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦) | |
11 | 9, 10 | sylib 220 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → ({𝑧} ∪ 𝑦) = 𝑦) |
12 | 8, 11 | syl5reqr 2874 | . . . . . 6 ⊢ (𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∪ {𝑧})) |
13 | vex 3500 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | eqvinc 3645 | . . . . . 6 ⊢ (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
15 | 12, 14 | sylib 220 | . . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
16 | 2 | bicomd 225 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
17 | 16, 3 | sylan9bb 512 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
18 | 17 | exlimiv 1930 | . . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
19 | 15, 18 | syl 17 | . . . 4 ⊢ (𝑧 ∈ 𝑦 → (𝜒 ↔ 𝜃)) |
20 | 19 | biimpd 231 | . . 3 ⊢ (𝑧 ∈ 𝑦 → (𝜒 → 𝜃)) |
21 | 7, 20 | pm2.61d2 183 | . 2 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
22 | 1, 2, 3, 4, 5, 21 | findcard2 8761 | 1 ⊢ (𝐴 ∈ Fin → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∪ cun 3937 ⊆ wss 3939 ∅c0 4294 {csn 4570 Fincfn 8512 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-1o 8105 df-er 8292 df-en 8513 df-fin 8516 |
This theorem is referenced by: findcard2d 8763 ac6sfi 8765 domunfican 8794 fodomfi 8800 hashxplem 13797 hashmap 13799 hashbc 13814 hashf1lem2 13817 hashf1 13818 fsum2d 15129 fsumabs 15159 fsumrlim 15169 fsumo1 15170 fsumiun 15179 incexclem 15194 fprod2d 15338 coprmprod 16008 coprmproddvds 16010 gsum2dlem2 19094 ablfac1eulem 19197 mplcoe1 20249 mplcoe5 20252 coe1fzgsumd 20473 evl1gsumd 20523 mdetunilem9 21232 ptcmpfi 22424 tmdgsum 22706 fsumcn 23481 ovolfiniun 24105 volfiniun 24151 itgfsum 24430 dvmptfsum 24575 jensen 25569 gsumle 30729 gsumvsca1 30858 gsumvsca2 30859 finixpnum 34881 matunitlindflem1 34892 pwslnm 39700 fnchoice 41292 dvmptfprod 42236 |
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