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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 9385), 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 9417. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 9384 | . . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
2 | 1onn 8254 | . . . . 5 ⊢ 1o ∈ ω | |
3 | cardnn 9380 | . . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1o) = 1o |
5 | 1, 4 | syl6eq 2869 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o) |
6 | 4 | eqeq2i 2831 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
7 | 6 | biimpri 229 | . . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o)) |
8 | 1n0 8108 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
9 | 8 | neii 3015 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
10 | eqeq1 2822 | . . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅)) | |
11 | 9, 10 | mtbiri 328 | . . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6693 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 143 | . . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card) |
14 | 1on 8098 | . . . . . 6 ⊢ 1o ∈ On | |
15 | onenon 9366 | . . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ dom card |
17 | carden2 9404 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) | |
18 | 13, 16, 17 | sylancl 586 | . . . 4 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) |
19 | 7, 18 | mpbid 233 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o) |
20 | 5, 19 | impbii 210 | . 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o) |
21 | 13 | elexd 3512 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ V) |
22 | fveqeq2 6672 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o)) | |
23 | 21, 22 | elab3 3671 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o) |
24 | 20, 23 | bitr4i 279 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cab 2796 ∅c0 4288 class class class wbr 5057 dom cdm 5548 Oncon0 6184 ‘cfv 6348 ωcom 7569 1oc1o 8084 ≈ cen 8494 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 |
This theorem is referenced by: (None) |
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