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Mirrors > Home > MPE Home > Th. List > ackbij1lem13 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9262. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
Ref | Expression |
---|---|
ackbij1lem13 | ⊢ (𝐹‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackbij.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) | |
2 | 1 | ackbij1lem10 9253 | . . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | peano1 7232 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | f0cli 6513 | . . . 4 ⊢ (𝐹‘∅) ∈ ω |
5 | nna0 7838 | . . . 4 ⊢ ((𝐹‘∅) ∈ ω → ((𝐹‘∅) +𝑜 ∅) = (𝐹‘∅)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝐹‘∅) +𝑜 ∅) = (𝐹‘∅) |
7 | un0 4111 | . . . 4 ⊢ (∅ ∪ ∅) = ∅ | |
8 | 7 | fveq2i 6335 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = (𝐹‘∅) |
9 | ackbij1lem3 9246 | . . . . 5 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin)) | |
10 | 3, 9 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ (𝒫 ω ∩ Fin) |
11 | in0 4112 | . . . 4 ⊢ (∅ ∩ ∅) = ∅ | |
12 | 1 | ackbij1lem9 9252 | . . . 4 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ ∅ ∈ (𝒫 ω ∩ Fin) ∧ (∅ ∩ ∅) = ∅) → (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +𝑜 (𝐹‘∅))) |
13 | 10, 10, 11, 12 | mp3an 1572 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +𝑜 (𝐹‘∅)) |
14 | 6, 8, 13 | 3eqtr2ri 2800 | . 2 ⊢ ((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) |
15 | nnacan 7862 | . . 3 ⊢ (((𝐹‘∅) ∈ ω ∧ (𝐹‘∅) ∈ ω ∧ ∅ ∈ ω) → (((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) ↔ (𝐹‘∅) = ∅)) | |
16 | 4, 4, 3, 15 | mp3an 1572 | . 2 ⊢ (((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) ↔ (𝐹‘∅) = ∅) |
17 | 14, 16 | mpbi 220 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∪ cun 3721 ∩ cin 3722 ∅c0 4063 𝒫 cpw 4297 {csn 4316 ∪ ciun 4654 ↦ cmpt 4863 × cxp 5247 ‘cfv 6031 (class class class)co 6793 ωcom 7212 +𝑜 coa 7710 Fincfn 8109 cardccrd 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8965 df-cda 9192 |
This theorem is referenced by: ackbij1lem14 9257 ackbij1 9262 |
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