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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 10006), 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 10039. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 10005 | . . . 4 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
2 | 1onn 8677 | . . . . 5 ⊢ 1o ∈ ω | |
3 | cardnn 10001 | . . . . 5 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1o) = 1o |
5 | 1, 4 | eqtrdi 2791 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = 1o) |
6 | 4 | eqeq2i 2748 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
7 | 6 | biimpri 228 | . . . 4 ⊢ ((card‘𝐴) = 1o → (card‘𝐴) = (card‘1o)) |
8 | 1n0 8525 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
9 | 8 | neii 2940 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
10 | eqeq1 2739 | . . . . . . 7 ⊢ ((card‘𝐴) = 1o → ((card‘𝐴) = ∅ ↔ 1o = ∅)) | |
11 | 9, 10 | mtbiri 327 | . . . . . 6 ⊢ ((card‘𝐴) = 1o → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6942 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 141 | . . . . 5 ⊢ ((card‘𝐴) = 1o → 𝐴 ∈ dom card) |
14 | 1on 8517 | . . . . . 6 ⊢ 1o ∈ On | |
15 | onenon 9987 | . . . . . 6 ⊢ (1o ∈ On → 1o ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ dom card |
17 | carden2 10025 | . . . . 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 6916 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1o ↔ (card‘𝐴) = 1o)) | |
22 | 13, 21 | elab3 3689 | . 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 1537 ∈ wcel 2106 {cab 2712 ∅c0 4339 class class class wbr 5148 dom cdm 5689 Oncon0 6386 ‘cfv 6563 ωcom 7887 1oc1o 8498 ≈ cen 8981 cardccrd 9973 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 |
This theorem is referenced by: (None) |
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