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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 10008), 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 10041. (Contributed by NM, 4-Nov-2013.) |
| Ref | Expression |
|---|---|
| pm54.43lem | ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | carden2b 10007 | . . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
| 2 | 1onn 8678 | . . . . 5 ⊢ 1o ∈ ω | |
| 3 | cardnn 10003 | . . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1o) = 1o |
| 5 | 1, 4 | eqtrdi 2793 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o) |
| 6 | 4 | eqeq2i 2750 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
| 7 | 6 | biimpri 228 | . . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o)) |
| 8 | 1n0 8526 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
| 9 | 8 | neii 2942 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
| 10 | eqeq1 2741 | . . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅)) | |
| 11 | 9, 10 | mtbiri 327 | . . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅) |
| 12 | ndmfv 6941 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
| 13 | 11, 12 | nsyl2 141 | . . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card) |
| 14 | 1on 8518 | . . . . . 6 ⊢ 1o ∈ On | |
| 15 | onenon 9989 | . . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card) | |
| 16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ dom card |
| 17 | carden2 10027 | . . . . 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 232 | . . 3 ⊢ ((card‘𝐴) = 1o → 𝐴 ≈ 1o) |
| 20 | 5, 19 | impbii 209 | . 2 ⊢ (𝐴 ≈ 1o ↔ (card‘𝐴) = 1o) |
| 21 | fveqeq2 6915 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o)) | |
| 22 | 13, 21 | elab3 3686 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} ↔ (card‘𝐴) = 1o) |
| 23 | 20, 22 | bitr4i 278 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 ∅c0 4333 class class class wbr 5143 dom cdm 5685 Oncon0 6384 ‘cfv 6561 ωcom 7887 1oc1o 8499 ≈ cen 8982 cardccrd 9975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 |
| This theorem is referenced by: (None) |
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