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Mirrors > Home > MPE Home > Th. List > ackbij1lem13 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 8943. (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 8934 | . . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | peano1 6977 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | f0cli 6278 | . . . 4 ⊢ (𝐹‘∅) ∈ ω |
5 | nna0 7571 | . . . 4 ⊢ ((𝐹‘∅) ∈ ω → ((𝐹‘∅) +𝑜 ∅) = (𝐹‘∅)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝐹‘∅) +𝑜 ∅) = (𝐹‘∅) |
7 | un0 3919 | . . . 4 ⊢ (∅ ∪ ∅) = ∅ | |
8 | 7 | fveq2i 6106 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = (𝐹‘∅) |
9 | ackbij1lem3 8927 | . . . . 5 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin)) | |
10 | 3, 9 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ (𝒫 ω ∩ Fin) |
11 | in0 3920 | . . . 4 ⊢ (∅ ∩ ∅) = ∅ | |
12 | 1 | ackbij1lem9 8933 | . . . 4 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ ∅ ∈ (𝒫 ω ∩ Fin) ∧ (∅ ∩ ∅) = ∅) → (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +𝑜 (𝐹‘∅))) |
13 | 10, 10, 11, 12 | mp3an 1416 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +𝑜 (𝐹‘∅)) |
14 | 6, 8, 13 | 3eqtr2ri 2639 | . 2 ⊢ ((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) |
15 | nnacan 7595 | . . 3 ⊢ (((𝐹‘∅) ∈ ω ∧ (𝐹‘∅) ∈ ω ∧ ∅ ∈ ω) → (((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) ↔ (𝐹‘∅) = ∅)) | |
16 | 4, 4, 3, 15 | mp3an 1416 | . 2 ⊢ (((𝐹‘∅) +𝑜 (𝐹‘∅)) = ((𝐹‘∅) +𝑜 ∅) ↔ (𝐹‘∅) = ∅) |
17 | 14, 16 | mpbi 219 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ωcom 6957 +𝑜 coa 7444 Fincfn 7841 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 |
This theorem is referenced by: ackbij1lem14 8938 ackbij1 8943 |
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