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Mirrors > Home > MPE Home > Th. List > card2inf | Structured version Visualization version GIF version |
Description: The alternate definition of the cardinal of a set given in cardval2 9934 has the curious property that for non-numerable sets (for which ndmfv 6882 yields ∅), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
card2inf.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
card2inf | ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5113 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ≺ 𝐴 ↔ ∅ ≺ 𝐴)) | |
2 | breq1 5113 | . . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴)) | |
3 | breq1 5113 | . . . . 5 ⊢ (𝑥 = suc 𝑛 → (𝑥 ≺ 𝐴 ↔ suc 𝑛 ≺ 𝐴)) | |
4 | 0elon 6376 | . . . . . . . 8 ⊢ ∅ ∈ On | |
5 | breq1 5113 | . . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 ≈ 𝐴 ↔ ∅ ≈ 𝐴)) | |
6 | 5 | rspcev 3584 | . . . . . . . 8 ⊢ ((∅ ∈ On ∧ ∅ ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
7 | 4, 6 | mpan 689 | . . . . . . 7 ⊢ (∅ ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
8 | 7 | con3i 154 | . . . . . 6 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ¬ ∅ ≈ 𝐴) |
9 | card2inf.1 | . . . . . . . 8 ⊢ 𝐴 ∈ V | |
10 | 9 | 0dom 9057 | . . . . . . 7 ⊢ ∅ ≼ 𝐴 |
11 | brsdom 8922 | . . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ (∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴)) | |
12 | 10, 11 | mpbiran 708 | . . . . . 6 ⊢ (∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴) |
13 | 8, 12 | sylibr 233 | . . . . 5 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∅ ≺ 𝐴) |
14 | sucdom2 9157 | . . . . . . . 8 ⊢ (𝑛 ≺ 𝐴 → suc 𝑛 ≼ 𝐴) | |
15 | 14 | ad2antll 728 | . . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≼ 𝐴) |
16 | nnon 7813 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On) | |
17 | onsuc 7751 | . . . . . . . . . 10 ⊢ (𝑛 ∈ On → suc 𝑛 ∈ On) | |
18 | breq1 5113 | . . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑛 → (𝑦 ≈ 𝐴 ↔ suc 𝑛 ≈ 𝐴)) | |
19 | 18 | rspcev 3584 | . . . . . . . . . . 11 ⊢ ((suc 𝑛 ∈ On ∧ suc 𝑛 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴) |
20 | 19 | ex 414 | . . . . . . . . . 10 ⊢ (suc 𝑛 ∈ On → (suc 𝑛 ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)) |
21 | 16, 17, 20 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑛 ∈ ω → (suc 𝑛 ≈ 𝐴 → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)) |
22 | 21 | con3dimp 410 | . . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴) → ¬ suc 𝑛 ≈ 𝐴) |
23 | 22 | adantrr 716 | . . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → ¬ suc 𝑛 ≈ 𝐴) |
24 | brsdom 8922 | . . . . . . 7 ⊢ (suc 𝑛 ≺ 𝐴 ↔ (suc 𝑛 ≼ 𝐴 ∧ ¬ suc 𝑛 ≈ 𝐴)) | |
25 | 15, 23, 24 | sylanbrc 584 | . . . . . 6 ⊢ ((𝑛 ∈ ω ∧ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 ∧ 𝑛 ≺ 𝐴)) → suc 𝑛 ≺ 𝐴) |
26 | 25 | exp32 422 | . . . . 5 ⊢ (𝑛 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑛 ≺ 𝐴 → suc 𝑛 ≺ 𝐴))) |
27 | 1, 2, 3, 13, 26 | finds2 7842 | . . . 4 ⊢ (𝑥 ∈ ω → (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → 𝑥 ≺ 𝐴)) |
28 | 27 | com12 32 | . . 3 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (𝑥 ∈ ω → 𝑥 ≺ 𝐴)) |
29 | 28 | ralrimiv 3143 | . 2 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∀𝑥 ∈ ω 𝑥 ≺ 𝐴) |
30 | omsson 7811 | . . 3 ⊢ ω ⊆ On | |
31 | ssrab 4035 | . . 3 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴)) | |
32 | 30, 31 | mpbiran 708 | . 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ↔ ∀𝑥 ∈ ω 𝑥 ≺ 𝐴) |
33 | 29, 32 | sylibr 233 | 1 ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 {crab 3410 Vcvv 3448 ⊆ wss 3915 ∅c0 4287 class class class wbr 5110 Oncon0 6322 suc csuc 6324 ωcom 7807 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: (None) |
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