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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 7502 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ On) | |
2 | oaword1 7519 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝑥)) | |
3 | ontri1 5674 | . . . . . 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 221 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) |
6 | 1, 5 | eldifd 3551 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴)) |
7 | 6 | ralrimiva 2949 | . 2 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ On (𝐴 +𝑜 𝑥) ∈ (On ∖ 𝐴)) |
8 | simpl 472 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ∈ On) | |
9 | eldifi 3694 | . . . . . 6 ⊢ (𝑦 ∈ (On ∖ 𝐴) → 𝑦 ∈ On) | |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝑦 ∈ On) |
11 | eldifn 3695 | . . . . . . 7 ⊢ (𝑦 ∈ (On ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴) |
13 | ontri1 5674 | . . . . . . 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 246 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → 𝐴 ⊆ 𝑦) |
16 | oawordeu 7522 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴 ⊆ 𝑦) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦) | |
17 | 8, 10, 15, 16 | syl21anc 1317 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦) |
18 | eqcom 2617 | . . . . 5 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +𝑜 𝑥)) | |
19 | 18 | reubii 3105 | . . . 4 ⊢ (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝑦 ↔ ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
20 | 17, 19 | sylib 207 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ (On ∖ 𝐴)) → ∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
21 | 20 | ralrimiva 2949 | . 2 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ (On ∖ 𝐴)∃!𝑥 ∈ On 𝑦 = (𝐴 +𝑜 𝑥)) |
22 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ On ↦ (𝐴 +𝑜 𝑥)) | |
23 | 22 | f1ompt 6290 | . 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 695 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 ∖ cdif 3537 ⊆ wss 3540 ↦ cmpt 4643 Oncon0 5640 –1-1-onto→wf1o 5803 (class class class)co 6549 +𝑜 coa 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 |
This theorem is referenced by: oacomf1olem 7531 |
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