![]() |
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 5320 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ∈ V) | |
2 | limenpsi.1 | . . . . . . 7 ⊢ Lim 𝐴 | |
3 | limsuc 7835 | . . . . . . 7 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐴) |
5 | 4 | biimpi 215 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) |
6 | nsuceq0 6441 | . . . . 5 ⊢ suc 𝑥 ≠ ∅ | |
7 | eldifsn 4785 | . . . . 5 ⊢ (suc 𝑥 ∈ (𝐴 ∖ {∅}) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ≠ ∅)) | |
8 | 5, 6, 7 | sylanblrc 589 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ (𝐴 ∖ {∅})) |
9 | limord 6418 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
10 | 2, 9 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
11 | ordelon 6382 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
12 | 10, 11 | mpan 687 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On) |
13 | ordelon 6382 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) | |
14 | 10, 13 | mpan 687 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ On) |
15 | suc11 6465 | . . . . 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 8994 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ {∅}) ∈ V) → 𝐴 ≼ (𝐴 ∖ {∅})) |
18 | 1, 17 | mpdan 684 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ∖ {∅})) |
19 | difss 4126 | . . 3 ⊢ (𝐴 ∖ {∅}) ⊆ 𝐴 | |
20 | ssdomg 8998 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ⊆ 𝐴 → (𝐴 ∖ {∅}) ≼ 𝐴)) | |
21 | 19, 20 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≼ 𝐴) |
22 | sbth 9095 | . 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 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 {csn 4623 class class class wbr 5141 Ord word 6357 Oncon0 6358 Lim wlim 6359 suc csuc 6360 ≈ cen 8938 ≼ cdom 8939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-en 8942 df-dom 8943 |
This theorem is referenced by: limensuci 9155 omenps 9652 infdifsn 9654 ominf4 10309 |
Copyright terms: Public domain | W3C validator |