![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5347 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ∈ V) | |
2 | limenpsi.1 | . . . . . . 7 ⊢ Lim 𝐴 | |
3 | limsuc 7886 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴) |
5 | 4 | biimpi 216 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) |
6 | nsuceq0 6478 | . . . . 5 ⊢ suc 𝑥 ≠ ∅ | |
7 | eldifsn 4811 | . . . . 5 ⊢ (suc 𝑥 ∈ (𝐴 ∖ {∅}) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) | |
8 | 5, 6, 7 | sylanblrc 589 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ (𝐴 ∖ {∅})) |
9 | limord 6455 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
10 | 2, 9 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
11 | ordelon 6419 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
12 | 10, 11 | mpan 689 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On) |
13 | ordelon 6419 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) | |
14 | 10, 13 | mpan 689 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ On) |
15 | suc11 6502 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) | |
16 | 12, 14, 15 | syl2an 595 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (suc 𝑥 = suc 𝑦 ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | dom3 9056 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ {∅}) ∈ V) → 𝐴 ≼ (𝐴 ∖ {∅})) |
18 | 1, 17 | mpdan 686 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ∖ {∅})) |
19 | difss 4159 | . . 3 ⊢ (𝐴 ∖ {∅}) ⊆ 𝐴 | |
20 | ssdomg 9060 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ⊆ 𝐴 → (𝐴 ∖ {∅}) ≼ 𝐴)) | |
21 | 19, 20 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≼ 𝐴) |
22 | sbth 9159 | . 2 ⊢ ((𝐴 ≼ (𝐴 ∖ {∅}) ∧ (𝐴 ∖ {∅}) ≼ 𝐴) → 𝐴 ≈ (𝐴 ∖ {∅})) | |
23 | 18, 21, 22 | syl2anc 583 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 ∅c0 4352 {csn 4648 class class class wbr 5166 Ord word 6394 Oncon0 6395 Lim wlim 6396 suc csuc 6397 ≈ cen 9000 ≼ cdom 9001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-en 9004 df-dom 9005 |
This theorem is referenced by: limensuci 9219 omenps 9724 infdifsn 9726 ominf4 10381 |
Copyright terms: Public domain | W3C validator |