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Mirrors > Home > MPE Home > Th. List > ackbij1lem13 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9456. (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 9447 | . . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | peano1 7414 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | f0cli 6685 | . . . 4 ⊢ (𝐹‘∅) ∈ ω |
5 | nna0 8029 | . . . 4 ⊢ ((𝐹‘∅) ∈ ω → ((𝐹‘∅) +o ∅) = (𝐹‘∅)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝐹‘∅) +o ∅) = (𝐹‘∅) |
7 | un0 4224 | . . . 4 ⊢ (∅ ∪ ∅) = ∅ | |
8 | 7 | fveq2i 6499 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = (𝐹‘∅) |
9 | ackbij1lem3 9440 | . . . . 5 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin)) | |
10 | 3, 9 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ (𝒫 ω ∩ Fin) |
11 | in0 4225 | . . . 4 ⊢ (∅ ∩ ∅) = ∅ | |
12 | 1 | ackbij1lem9 9446 | . . . 4 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ ∅ ∈ (𝒫 ω ∩ Fin) ∧ (∅ ∩ ∅) = ∅) → (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅))) |
13 | 10, 10, 11, 12 | mp3an 1441 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅)) |
14 | 6, 8, 13 | 3eqtr2ri 2802 | . 2 ⊢ ((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) |
15 | nnacan 8053 | . . 3 ⊢ (((𝐹‘∅) ∈ ω ∧ (𝐹‘∅) ∈ ω ∧ ∅ ∈ ω) → (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅)) | |
16 | 4, 4, 3, 15 | mp3an 1441 | . 2 ⊢ (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅) |
17 | 14, 16 | mpbi 222 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 ∈ wcel 2051 ∪ cun 3820 ∩ cin 3821 ∅c0 4172 𝒫 cpw 4416 {csn 4435 ∪ ciun 4788 ↦ cmpt 5004 × cxp 5401 ‘cfv 6185 (class class class)co 6974 ωcom 7394 +o coa 7900 Fincfn 8304 cardccrd 9156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 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 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 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-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 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-pred 5983 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-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-dju 9122 df-card 9160 |
This theorem is referenced by: ackbij1lem14 9451 ackbij1 9456 |
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