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Mirrors > Home > MPE Home > Th. List > findcard2s | Structured version Visualization version GIF version |
Description: Variation of findcard2 9111 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 414 | . . 3 ⊢ (𝑦 ∈ Fin → (¬ 𝑧 ∈ 𝑦 → (𝜒 → 𝜃))) |
8 | snssi 4769 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → {𝑧} ⊆ 𝑦) | |
9 | ssequn1 4141 | . . . . . . . 8 ⊢ ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦) | |
10 | 8, 9 | sylib 217 | . . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → ({𝑧} ∪ 𝑦) = 𝑦) |
11 | uncom 4114 | . . . . . . 7 ⊢ ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧}) | |
12 | 10, 11 | eqtr3di 2788 | . . . . . 6 ⊢ (𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∪ {𝑧})) |
13 | vex 3448 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
14 | 13 | eqvinc 3600 | . . . . . 6 ⊢ (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
15 | 12, 14 | sylib 217 | . . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧}))) |
16 | 2 | bicomd 222 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
17 | 16, 3 | sylan9bb 511 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
18 | 17 | exlimiv 1934 | . . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = (𝑦 ∪ {𝑧})) → (𝜒 ↔ 𝜃)) |
19 | 15, 18 | syl 17 | . . . 4 ⊢ (𝑧 ∈ 𝑦 → (𝜒 ↔ 𝜃)) |
20 | 19 | biimpd 228 | . . 3 ⊢ (𝑧 ∈ 𝑦 → (𝜒 → 𝜃)) |
21 | 7, 20 | pm2.61d2 181 | . 2 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
22 | 1, 2, 3, 4, 5, 21 | findcard2 9111 | 1 ⊢ (𝐴 ∈ Fin → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∪ cun 3909 ⊆ wss 3911 ∅c0 4283 {csn 4587 Fincfn 8886 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-en 8887 df-fin 8890 |
This theorem is referenced by: findcard2d 9113 unfi 9119 ac6sfi 9234 domunfican 9267 fodomfi 9272 hashxplem 14339 hashmap 14341 hashbc 14356 hashf1lem2 14361 hashf1 14362 fsum2d 15661 fsumabs 15691 fsumrlim 15701 fsumo1 15702 fsumiun 15711 incexclem 15726 fprod2d 15869 coprmprod 16542 coprmproddvds 16544 gsum2dlem2 19753 ablfac1eulem 19856 mplcoe1 21454 mplcoe5 21457 coe1fzgsumd 21689 evl1gsumd 21739 mdetunilem9 21985 ptcmpfi 23180 tmdgsum 23462 fsumcn 24249 ovolfiniun 24881 volfiniun 24927 itgfsum 25207 dvmptfsum 25355 jensen 26354 gsumle 31981 gsumvsca1 32110 gsumvsca2 32111 finixpnum 36109 matunitlindflem1 36120 pwslnm 41464 fnchoice 43322 dvmptfprod 44272 |
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