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Mirrors > Home > MPE Home > Th. List > pm54.43lem | Structured version Visualization version GIF version |
Description: In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9396), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}. Here we show that this is equivalent to 𝐴 ≈ 1o so that we can use the latter more convenient notation in pm54.43 9428. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 9395 | . . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
2 | 1onn 8264 | . . . . 5 ⊢ 1o ∈ ω | |
3 | cardnn 9391 | . . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1o) = 1o |
5 | 1, 4 | syl6eq 2872 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o) |
6 | 4 | eqeq2i 2834 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
7 | 6 | biimpri 230 | . . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o)) |
8 | 1n0 8118 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
9 | 8 | neii 3018 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
10 | eqeq1 2825 | . . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅)) | |
11 | 9, 10 | mtbiri 329 | . . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6699 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 143 | . . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card) |
14 | 1on 8108 | . . . . . 6 ⊢ 1o ∈ On | |
15 | onenon 9377 | . . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ dom card |
17 | carden2 9415 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) | |
18 | 13, 16, 17 | sylancl 588 | . . . 4 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) |
19 | 7, 18 | mpbid 234 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o) |
20 | 5, 19 | impbii 211 | . 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o) |
21 | 13 | elexd 3514 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ V) |
22 | fveqeq2 6678 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o)) | |
23 | 21, 22 | elab3 3673 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o) |
24 | 20, 23 | bitr4i 280 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 ∅c0 4290 class class class wbr 5065 dom cdm 5554 Oncon0 6190 ‘cfv 6354 ωcom 7579 1oc1o 8094 ≈ cen 8505 cardccrd 9363 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 |
This theorem is referenced by: (None) |
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