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Mirrors > Home > MPE Home > Th. List > findcard2s | Structured version Visualization version GIF version |
Description: Variation of findcard2 8551 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 405 | . . 3 ⊢ (𝑦 ∈ Fin → (¬ 𝑧 ∈ 𝑦 → (𝜒 → 𝜃))) |
8 | uncom 4012 | . . . . . . 7 ⊢ ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧}) | |
9 | snssi 4611 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦) | |
10 | ssequn1 4038 | . . . . . . . 8 ⊢ ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦) | |
11 | 9, 10 | sylib 210 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → ({𝑧} ∪ 𝑦) = 𝑦) |
12 | 8, 11 | syl5reqr 2823 | . . . . . 6 ⊢ (𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∪ {𝑧})) |
13 | vex 3412 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | eqvinc 3551 | . . . . . 6 ⊢ (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
15 | 12, 14 | sylib 210 | . . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
16 | 2 | bicomd 215 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
17 | 16, 3 | sylan9bb 502 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
18 | 17 | exlimiv 1889 | . . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
19 | 15, 18 | syl 17 | . . . 4 ⊢ (𝑧 ∈ 𝑦 → (𝜒 ↔ 𝜃)) |
20 | 19 | biimpd 221 | . . 3 ⊢ (𝑧 ∈ 𝑦 → (𝜒 → 𝜃)) |
21 | 7, 20 | pm2.61d2 174 | . 2 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
22 | 1, 2, 3, 4, 5, 21 | findcard2 8551 | 1 ⊢ (𝐴 ∈ Fin → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∪ cun 3821 ⊆ wss 3823 ∅c0 4172 {csn 4435 Fincfn 8304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-1o 7903 df-er 8087 df-en 8305 df-fin 8308 |
This theorem is referenced by: findcard2d 8553 ac6sfi 8555 domunfican 8584 fodomfi 8590 hashxplem 13605 hashmap 13607 hashbc 13622 hashf1lem2 13625 hashf1 13626 fsum2d 14984 fsumabs 15014 fsumrlim 15024 fsumo1 15025 fsumiun 15034 incexclem 15049 fprod2d 15193 coprmprod 15859 coprmproddvds 15861 gsum2dlem2 18856 ablfac1eulem 18956 mplcoe1 19971 mplcoe5 19974 coe1fzgsumd 20185 evl1gsumd 20234 mdetunilem9 20945 ptcmpfi 22137 tmdgsum 22419 fsumcn 23193 ovolfiniun 23817 volfiniun 23863 itgfsum 24142 dvmptfsum 24287 jensen 25280 gsumle 30551 gsumvsca1 30554 gsumvsca2 30555 finixpnum 34347 matunitlindflem1 34358 pwslnm 39119 fnchoice 40734 dvmptfprod 41685 |
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