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| Mirrors > Home > MPE Home > Th. List > limensuci | Structured version Visualization version GIF version | ||
| Description: A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
| Ref | Expression |
|---|---|
| limensuci.1 | ⊢ Lim 𝐴 |
| Ref | Expression |
|---|---|
| limensuci | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | limensuci.1 | . . . . 5 ⊢ Lim 𝐴 | |
| 2 | 1 | limenpsi 9192 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) |
| 3 | 2 | ensymd 9045 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ {∅}) ≈ 𝐴) |
| 4 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 5 | en2sn 9081 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → {∅} ≈ {𝐴}) | |
| 6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {∅} ≈ {𝐴}) |
| 7 | disjdifr 4473 | . . . 4 ⊢ ((𝐴 ∖ {∅}) ∩ {∅}) = ∅ | |
| 8 | limord 6444 | . . . . . . 7 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 9 | 1, 8 | ax-mp 5 | . . . . . 6 ⊢ Ord 𝐴 |
| 10 | ordirr 6402 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
| 11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ ¬ 𝐴 ∈ 𝐴 |
| 12 | disjsn 4711 | . . . . 5 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
| 13 | 11, 12 | mpbir 231 | . . . 4 ⊢ (𝐴 ∩ {𝐴}) = ∅ |
| 14 | unen 9086 | . . . 4 ⊢ ((((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) ∧ (((𝐴 ∖ {∅}) ∩ {∅}) = ∅ ∧ (𝐴 ∩ {𝐴}) = ∅)) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) | |
| 15 | 7, 13, 14 | mpanr12 705 | . . 3 ⊢ (((𝐴 ∖ {∅}) ≈ 𝐴 ∧ {∅} ≈ {𝐴}) → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
| 16 | 3, 6, 15 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ {∅}) ∪ {∅}) ≈ (𝐴 ∪ {𝐴})) |
| 17 | 0ellim 6447 | . . . . . 6 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) | |
| 18 | 1, 17 | ax-mp 5 | . . . . 5 ⊢ ∅ ∈ 𝐴 |
| 19 | 4 | snss 4785 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ {∅} ⊆ 𝐴) |
| 20 | 18, 19 | mpbi 230 | . . . 4 ⊢ {∅} ⊆ 𝐴 |
| 21 | undif 4482 | . . . 4 ⊢ ({∅} ⊆ 𝐴 ↔ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴) | |
| 22 | 20, 21 | mpbi 230 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = 𝐴 |
| 23 | uncom 4158 | . . 3 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ((𝐴 ∖ {∅}) ∪ {∅}) | |
| 24 | 22, 23 | eqtr3i 2767 | . 2 ⊢ 𝐴 = ((𝐴 ∖ {∅}) ∪ {∅}) |
| 25 | df-suc 6390 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 26 | 16, 24, 25 | 3brtr4g 5177 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 Ord word 6383 Lim wlim 6385 suc csuc 6386 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 |
| This theorem is referenced by: limensuc 9194 infensuc 9195 omensuc 9696 |
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