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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 8794), 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 8826. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 8793 | . . . 4 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜)) | |
2 | 1onn 7719 | . . . . 5 ⊢ 1𝑜 ∈ ω | |
3 | cardnn 8789 | . . . . 5 ⊢ (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1𝑜) = 1𝑜 |
5 | 1, 4 | syl6eq 2672 | . . 3 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜) |
6 | 4 | eqeq2i 2634 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜) |
7 | 6 | biimpri 218 | . . . 4 ⊢ ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜)) |
8 | 1n0 7575 | . . . . . . . 8 ⊢ 1𝑜 ≠ ∅ | |
9 | 8 | neii 2796 | . . . . . . 7 ⊢ ¬ 1𝑜 = ∅ |
10 | eqeq1 2626 | . . . . . . 7 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅)) | |
11 | 9, 10 | mtbiri 317 | . . . . . 6 ⊢ ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6218 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 142 | . . . . 5 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ dom card) |
14 | 1on 7567 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
15 | onenon 8775 | . . . . . 6 ⊢ (1𝑜 ∈ On → 1𝑜 ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1𝑜 ∈ dom card |
17 | carden2 8813 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜)) | |
18 | 13, 16, 17 | sylancl 694 | . . . 4 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜)) |
19 | 7, 18 | mpbid 222 | . . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ≈ 1𝑜) |
20 | 5, 19 | impbii 199 | . 2 ⊢ (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜) |
21 | elex 3212 | . . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V) | |
22 | 13, 21 | syl 17 | . . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ V) |
23 | fveq2 6191 | . . . 4 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
24 | 23 | eqeq1d 2624 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜)) |
25 | 22, 24 | elab3 3358 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜) |
26 | 20, 25 | bitr4i 267 | 1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 ∅c0 3915 class class class wbr 4653 dom cdm 5114 Oncon0 5723 ‘cfv 5888 ωcom 7065 1𝑜c1o 7553 ≈ cen 7952 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 |
This theorem is referenced by: (None) |
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