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Mirrors > Home > MPE Home > Th. List > fin1a2lem2 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 10452. 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 7830 | . . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
3 | 1, 2 | fmpti 7131 | . 2 ⊢ 𝑆:On⟶On |
4 | 1 | fin1a2lem1 10437 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎) |
5 | 1 | fin1a2lem1 10437 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
6 | 4, 5 | eqeqan12d 2748 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏)) |
7 | suc11 6492 | . . . . 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 3196 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏) |
11 | dff13 7274 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ↦ cmpt 5230 Oncon0 6385 suc csuc 6387 ⟶wf 6558 –1-1→wf1 6559 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fv 6570 |
This theorem is referenced by: fin1a2lem6 10442 onsucf1o 43261 |
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