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Mirrors > Home > MPE Home > Th. List > ackbij1lem13 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10275. (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 10266 | . . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | peano1 7911 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | f0cli 7118 | . . . 4 ⊢ (𝐹‘∅) ∈ ω |
5 | nna0 8641 | . . . 4 ⊢ ((𝐹‘∅) ∈ ω → ((𝐹‘∅) +o ∅) = (𝐹‘∅)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((𝐹‘∅) +o ∅) = (𝐹‘∅) |
7 | un0 4400 | . . . 4 ⊢ (∅ ∪ ∅) = ∅ | |
8 | 7 | fveq2i 6910 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = (𝐹‘∅) |
9 | ackbij1lem3 10259 | . . . . 5 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin)) | |
10 | 3, 9 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ (𝒫 ω ∩ Fin) |
11 | in0 4401 | . . . 4 ⊢ (∅ ∩ ∅) = ∅ | |
12 | 1 | ackbij1lem9 10265 | . . . 4 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ ∅ ∈ (𝒫 ω ∩ Fin) ∧ (∅ ∩ ∅) = ∅) → (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅))) |
13 | 10, 10, 11, 12 | mp3an 1460 | . . 3 ⊢ (𝐹‘(∅ ∪ ∅)) = ((𝐹‘∅) +o (𝐹‘∅)) |
14 | 6, 8, 13 | 3eqtr2ri 2770 | . 2 ⊢ ((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) |
15 | nnacan 8665 | . . 3 ⊢ (((𝐹‘∅) ∈ ω ∧ (𝐹‘∅) ∈ ω ∧ ∅ ∈ ω) → (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅)) | |
16 | 4, 4, 3, 15 | mp3an 1460 | . 2 ⊢ (((𝐹‘∅) +o (𝐹‘∅)) = ((𝐹‘∅) +o ∅) ↔ (𝐹‘∅) = ∅) |
17 | 14, 16 | mpbi 230 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 𝒫 cpw 4605 {csn 4631 ∪ ciun 4996 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ωcom 7887 +o coa 8502 Fincfn 8984 cardccrd 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 |
This theorem is referenced by: ackbij1lem14 10270 ackbij1 10275 |
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