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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 9998), 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 10031. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 9997 | . . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
2 | 1onn 8661 | . . . . 5 ⊢ 1o ∈ ω | |
3 | cardnn 9993 | . . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1o) = 1o |
5 | 1, 4 | eqtrdi 2781 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o) |
6 | 4 | eqeq2i 2738 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
7 | 6 | biimpri 227 | . . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o)) |
8 | 1n0 8509 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
9 | 8 | neii 2931 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
10 | eqeq1 2729 | . . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅)) | |
11 | 9, 10 | mtbiri 326 | . . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6931 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 141 | . . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card) |
14 | 1on 8499 | . . . . . 6 ⊢ 1o ∈ On | |
15 | onenon 9979 | . . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ dom card |
17 | carden2 10017 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) | |
18 | 13, 16, 17 | sylancl 584 | . . . 4 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)) |
19 | 7, 18 | mpbid 231 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o) |
20 | 5, 19 | impbii 208 | . 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o) |
21 | fveqeq2 6905 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o)) | |
22 | 13, 21 | elab3 3672 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o) |
23 | 20, 22 | bitr4i 277 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2702 ∅c0 4322 class class class wbr 5149 dom cdm 5678 Oncon0 6371 ‘cfv 6549 ωcom 7871 1oc1o 8480 ≈ cen 8961 cardccrd 9965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9969 |
This theorem is referenced by: (None) |
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