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| Mirrors > Home > MPE Home > Th. List > findcard2s | Structured version Visualization version GIF version | ||
| Description: Variation of findcard2 9204 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 412 | . . 3 ⊢ (𝑦 ∈ Fin → (¬ 𝑧 ∈ 𝑦 → (𝜒 → 𝜃))) |
| 8 | snssi 4808 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦) | |
| 9 | ssequn1 4186 | . . . . . . . 8 ⊢ ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦) | |
| 10 | 8, 9 | sylib 218 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → ({𝑧} ∪ 𝑦) = 𝑦) |
| 11 | uncom 4158 | . . . . . . 7 ⊢ ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧}) | |
| 12 | 10, 11 | eqtr3di 2792 | . . . . . 6 ⊢ (𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∪ {𝑧})) |
| 13 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 14 | 13 | eqvinc 3649 | . . . . . 6 ⊢ (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
| 15 | 12, 14 | sylib 218 | . . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
| 16 | 2 | bicomd 223 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
| 17 | 16, 3 | sylan9bb 509 | . . . . . 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 229 | . . 3 ⊢ (𝑧 ∈ 𝑦 → (𝜒 → 𝜃)) |
| 21 | 7, 20 | pm2.61d2 181 | . 2 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
| 22 | 1, 2, 3, 4, 5, 21 | findcard2 9204 | 1 ⊢ (𝐴 ∈ Fin → 𝜏) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 Fincfn 8985 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-en 8986 df-fin 8989 |
| This theorem is referenced by: findcard2d 9206 unfi 9211 ac6sfi 9320 fodomfi 9350 domunfican 9361 fodomfiOLD 9370 hashxplem 14472 hashmap 14474 hashbc 14492 hashf1lem2 14495 hashf1 14496 fsum2d 15807 fsumabs 15837 fsumrlim 15847 fsumo1 15848 fsumiun 15857 incexclem 15872 fprod2d 16017 coprmprod 16698 coprmproddvds 16700 gsum2dlem2 19989 ablfac1eulem 20092 mplcoe1 22055 mplcoe5 22058 coe1fzgsumd 22308 evl1gsumd 22361 mdetunilem9 22626 ptcmpfi 23821 tmdgsum 24103 fsumcn 24894 ovolfiniun 25536 volfiniun 25582 itgfsum 25862 dvmptfsum 26013 jensen 27032 gsumle 33101 gsumvsca1 33232 gsumvsca2 33233 finixpnum 37612 matunitlindflem1 37623 pwslnm 43106 fnchoice 45034 dvmptfprod 45960 |
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