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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 8677), 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 8709. (Contributed by NM, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm54.43lem | ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carden2b 8676 | . . . 4 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜)) | |
2 | 1onn 7606 | . . . . 5 ⊢ 1𝑜 ∈ ω | |
3 | cardnn 8672 | . . . . 5 ⊢ (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (card‘1𝑜) = 1𝑜 |
5 | 1, 4 | syl6eq 2660 | . . 3 ⊢ (𝐴 ≈ 1𝑜 → (card‘𝐴) = 1𝑜) |
6 | 4 | eqeq2i 2622 | . . . . 5 ⊢ ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜) |
7 | 6 | biimpri 217 | . . . 4 ⊢ ((card‘𝐴) = 1𝑜 → (card‘𝐴) = (card‘1𝑜)) |
8 | 1n0 7462 | . . . . . . . 8 ⊢ 1𝑜 ≠ ∅ | |
9 | 8 | neii 2784 | . . . . . . 7 ⊢ ¬ 1𝑜 = ∅ |
10 | eqeq1 2614 | . . . . . . 7 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = ∅ ↔ 1𝑜 = ∅)) | |
11 | 9, 10 | mtbiri 316 | . . . . . 6 ⊢ ((card‘𝐴) = 1𝑜 → ¬ (card‘𝐴) = ∅) |
12 | ndmfv 6128 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
13 | 11, 12 | nsyl2 141 | . . . . 5 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ dom card) |
14 | 1on 7454 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
15 | onenon 8658 | . . . . . 6 ⊢ (1𝑜 ∈ On → 1𝑜 ∈ dom card) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ 1𝑜 ∈ dom card |
17 | carden2 8696 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜)) | |
18 | 13, 16, 17 | sylancl 693 | . . . 4 ⊢ ((card‘𝐴) = 1𝑜 → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜)) |
19 | 7, 18 | mpbid 221 | . . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ≈ 1𝑜) |
20 | 5, 19 | impbii 198 | . 2 ⊢ (𝐴 ≈ 1𝑜 ↔ (card‘𝐴) = 1𝑜) |
21 | elex 3185 | . . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ∈ V) | |
22 | 13, 21 | syl 17 | . . 3 ⊢ ((card‘𝐴) = 1𝑜 → 𝐴 ∈ V) |
23 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝐴 → (card‘𝑥) = (card‘𝐴)) | |
24 | 23 | eqeq1d 2612 | . . 3 ⊢ (𝑥 = 𝐴 → ((card‘𝑥) = 1𝑜 ↔ (card‘𝐴) = 1𝑜)) |
25 | 22, 24 | elab3 3327 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} ↔ (card‘𝐴) = 1𝑜) |
26 | 20, 25 | bitr4i 266 | 1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∅c0 3874 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 ωcom 6957 1𝑜c1o 7440 ≈ cen 7838 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 |
This theorem is referenced by: (None) |
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