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| Mirrors > Home > MPE Home > Th. List > fin1a2lem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for fin1a2 10455. The successor operation on the ordinal numbers is injective or one-to-one. Lemma 1.17 of [Schloeder] p. 2. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
| Ref | Expression |
|---|---|
| fin1a2lem.a | ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
| Ref | Expression |
|---|---|
| fin1a2lem2 | ⊢ 𝑆:On–1-1→On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fin1a2lem.a | . . 3 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) | |
| 2 | onsuc 7831 | . . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
| 3 | 1, 2 | fmpti 7132 | . 2 ⊢ 𝑆:On⟶On |
| 4 | 1 | fin1a2lem1 10440 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎) |
| 5 | 1 | fin1a2lem1 10440 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
| 6 | 4, 5 | eqeqan12d 2751 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏)) |
| 7 | suc11 6491 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏)) | |
| 8 | 6, 7 | bitrd 279 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ 𝑎 = 𝑏)) |
| 9 | 8 | biimpd 229 | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)) |
| 10 | 9 | rgen2 3199 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏) |
| 11 | dff13 7275 | . 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏))) | |
| 12 | 3, 10, 11 | mpbir2an 711 | 1 ⊢ 𝑆:On–1-1→On |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 Oncon0 6384 suc csuc 6386 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 |
| 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-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-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-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 |
| This theorem is referenced by: fin1a2lem6 10445 onsucf1o 43285 |
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