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Mirrors > Home > MPE Home > Th. List > findcard2s | Structured version Visualization version GIF version |
Description: Variation of findcard2 9202 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 4812 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦) | |
9 | ssequn1 4195 | . . . . . . . 8 ⊢ ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦) | |
10 | 8, 9 | sylib 218 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → ({𝑧} ∪ 𝑦) = 𝑦) |
11 | uncom 4167 | . . . . . . 7 ⊢ ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧}) | |
12 | 10, 11 | eqtr3di 2789 | . . . . . 6 ⊢ (𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∪ {𝑧})) |
13 | vex 3481 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | eqvinc 3648 | . . . . . 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 1927 | . . . . 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 9202 | 1 ⊢ (𝐴 ∈ Fin → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 {csn 4630 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-en 8984 df-fin 8987 |
This theorem is referenced by: findcard2d 9204 unfi 9209 ac6sfi 9317 fodomfi 9347 domunfican 9358 fodomfiOLD 9367 hashxplem 14468 hashmap 14470 hashbc 14488 hashf1lem2 14491 hashf1 14492 fsum2d 15803 fsumabs 15833 fsumrlim 15843 fsumo1 15844 fsumiun 15853 incexclem 15868 fprod2d 16013 coprmprod 16694 coprmproddvds 16696 gsum2dlem2 20003 ablfac1eulem 20106 mplcoe1 22072 mplcoe5 22075 coe1fzgsumd 22323 evl1gsumd 22376 mdetunilem9 22641 ptcmpfi 23836 tmdgsum 24118 fsumcn 24907 ovolfiniun 25549 volfiniun 25595 itgfsum 25876 dvmptfsum 26027 jensen 27046 gsumle 33083 gsumvsca1 33214 gsumvsca2 33215 finixpnum 37591 matunitlindflem1 37602 pwslnm 43082 fnchoice 44966 dvmptfprod 45900 |
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