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Mirrors > Home > MPE Home > Th. List > oaf1o | Structured version Visualization version GIF version |
Description: Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.) |
Ref | Expression |
---|---|
oaf1o | ⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oacl 7660 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On) | |
2 | oaword1 7677 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝑥)) | |
3 | ontri1 5795 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐴 +𝑜 𝑥) ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)) | |
4 | 1, 3 | syldan 486 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)) |
5 | 2, 4 | mpbid 222 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) |
6 | 1, 5 | eldifd 3618 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴)) |
7 | 6 | ralrimiva 2995 | . 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴)) |
8 | simpl 472 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On) | |
9 | eldifi 3765 | . . . . . 6 ⊢ (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On) | |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On) |
11 | eldifn 3766 | . . . . . . 7 ⊢ (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴) |
13 | ontri1 5795 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴)) | |
14 | 10, 13 | syldan 486 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴)) |
15 | 12, 14 | mpbird 247 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ⊆ 𝑦) |
16 | oawordeu 7680 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝑦) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦) | |
17 | 8, 10, 15, 16 | syl21anc 1365 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦) |
18 | eqcom 2658 | . . . . 5 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +𝑜 𝑥)) | |
19 | 18 | reubii 3158 | . . . 4 ⊢ (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
20 | 17, 19 | sylib 208 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
21 | 20 | ralrimiva 2995 | . 2 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
22 | eqid 2651 | . . 3 ⊢ (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) | |
23 | 22 | f1ompt 6422 | . 2 ⊢ ((𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴) ↔ (∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴) ∧ ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥))) |
24 | 7, 21, 23 | sylanbrc 699 | 1 ⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)):On–1-1-onto→(On ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∃!wreu 2943 ∖ cdif 3604 ⊆ wss 3607 ↦ cmpt 4762 Oncon0 5761 –1-1-onto→wf1o 5925 (class class class)co 6690 +𝑜 coa 7602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-oadd 7609 |
This theorem is referenced by: oacomf1olem 7689 |
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