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Mirrors > Home > MPE Home > Th. List > limenpsi | Structured version Visualization version GIF version |
Description: A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
limenpsi.1 | ⊢ Lim 𝐴 |
Ref | Expression |
---|---|
limenpsi | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 5230 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ∈ V) | |
2 | limenpsi.1 | . . . . . . 7 ⊢ Lim 𝐴 | |
3 | limsuc 7563 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴) |
5 | 4 | biimpi 218 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) |
6 | nsuceq0 6270 | . . . . 5 ⊢ suc 𝑥 ≠ ∅ | |
7 | eldifsn 4718 | . . . . 5 ⊢ (suc 𝑥 ∈ (𝐴 ∖ {∅}) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) | |
8 | 5, 6, 7 | sylanblrc 592 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ (𝐴 ∖ {∅})) |
9 | limord 6249 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
10 | 2, 9 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
11 | ordelon 6214 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
12 | 10, 11 | mpan 688 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On) |
13 | ordelon 6214 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) | |
14 | 10, 13 | mpan 688 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ On) |
15 | suc11 6293 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) | |
16 | 12, 14, 15 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | dom3 8552 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ {∅}) ∈ V) → 𝐴 ≼ (𝐴 ∖ {∅})) |
18 | 1, 17 | mpdan 685 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ∖ {∅})) |
19 | difss 4107 | . . 3 ⊢ (𝐴 ∖ {∅}) ⊆ 𝐴 | |
20 | ssdomg 8554 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ⊆ 𝐴 → (𝐴 ∖ {∅}) ≼ 𝐴)) | |
21 | 19, 20 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≼ 𝐴) |
22 | sbth 8636 | . 2 ⊢ ((𝐴 ≼ (𝐴 ∖ {∅}) ∧ (𝐴 ∖ {∅}) ≼ 𝐴) → 𝐴 ≈ (𝐴 ∖ {∅})) | |
23 | 18, 21, 22 | syl2anc 586 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 {csn 4566 class class class wbr 5065 Ord word 6189 Oncon0 6190 Lim wlim 6191 suc csuc 6192 ≈ cen 8505 ≼ cdom 8506 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-en 8509 df-dom 8510 |
This theorem is referenced by: limensuci 8692 omenps 9117 infdifsn 9119 ominf4 9733 |
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