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Mirrors > Home > MPE Home > Th. List > oien | Structured version Visualization version GIF version |
Description: The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.) |
Ref | Expression |
---|---|
oicl.1 | ⊢ 𝐹 = OrdIso(𝑅, 𝐴) |
Ref | Expression |
---|---|
oien | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oicl.1 | . . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
2 | 1 | oiexg 8323 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 ∈ V) |
4 | 1 | oiiso 8325 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
5 | isof1o 6473 | . . 3 ⊢ (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) |
7 | f1oen3g 7857 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:dom 𝐹–1-1-onto→𝐴) → dom 𝐹 ≈ 𝐴) | |
8 | 3, 6, 7 | syl2anc 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 E cep 4947 We wwe 4996 dom cdm 5038 –1-1-onto→wf1o 5803 Isom wiso 5805 ≈ cen 7838 OrdIsocoi 8297 |
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-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-se 4998 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-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-en 7842 df-oi 8298 |
This theorem is referenced by: hartogslem1 8330 wofib 8333 cantnfcl 8447 cantnff 8454 cantnf0 8455 cantnfp1lem2 8459 cantnflem1 8469 cantnf 8473 cnfcom2lem 8481 finnisoeu 8819 dfac12lem2 8849 pwfseqlem5 9364 fz1isolem 13102 |
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