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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 | . . 3 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
2 | 1 | oiexg 9479 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
3 | 1 | oiiso 9481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
4 | isof1o 7272 | . . 3 ⊢ (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) |
6 | f1oen3g 8912 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:dom 𝐹–1-1-onto→𝐴) → dom 𝐹 ≈ 𝐴) | |
7 | 2, 5, 6 | syl2an2r 684 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 class class class wbr 5109 E cep 5540 We wwe 5591 dom cdm 5637 –1-1-onto→wf1o 6499 Isom wiso 6501 ≈ cen 8886 OrdIsocoi 9453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-en 8890 df-oi 9454 |
This theorem is referenced by: hartogslem1 9486 wofib 9489 cantnfcl 9611 cantnff 9618 cantnf0 9619 cantnfp1lem2 9623 cantnflem1 9633 cantnf 9637 cnfcom2lem 9645 finnisoeu 10057 dfac12lem2 10088 pwfseqlem5 10607 fz1isolem 14369 |
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