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Mirrors > Home > MPE Home > Th. List > ackbij1lem13 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9835. (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 9826 | . . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | peano1 7656 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | f0cli 6906 | . . . 4 ⊢ (𝐹‘∅) ∈ ω |
5 | nna0 8321 | . . . 4 ⊢ ((𝐹‘∅) ∈ ω → ((𝐹‘∅) +o ∅) = (𝐹‘∅)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝐹‘∅) +o ∅) = (𝐹‘∅) |
7 | un0 4295 | . . . 4 ⊢ (∅ ∪ ∅) = ∅ | |
8 | 7 | fveq2i 6709 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = (𝐹‘∅) |
9 | ackbij1lem3 9819 | . . . . 5 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin)) | |
10 | 3, 9 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ (𝒫 ω ∩ Fin) |
11 | in0 4296 | . . . 4 ⊢ (∅ ∩ ∅) = ∅ | |
12 | 1 | ackbij1lem9 9825 | . . . 4 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ ∅ ∈ (𝒫 ω ∩ Fin) ∧ (∅ ∩ ∅) = ∅) → (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅))) |
13 | 10, 10, 11, 12 | mp3an 1463 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅)) |
14 | 6, 8, 13 | 3eqtr2ri 2769 | . 2 ⊢ ((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) |
15 | nnacan 8345 | . . 3 ⊢ (((𝐹‘∅) ∈ ω ∧ (𝐹‘∅) ∈ ω ∧ ∅ ∈ ω) → (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅)) | |
16 | 4, 4, 3, 15 | mp3an 1463 | . 2 ⊢ (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅) |
17 | 14, 16 | mpbi 233 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∪ cun 3855 ∩ cin 3856 ∅c0 4227 𝒫 cpw 4503 {csn 4531 ∪ ciun 4894 ↦ cmpt 5124 × cxp 5538 ‘cfv 6369 (class class class)co 7202 ωcom 7633 +o coa 8188 Fincfn 8615 cardccrd 9534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 |
This theorem is referenced by: ackbij1lem14 9830 ackbij1 9835 |
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