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Mirrors > Home > MPE Home > Th. List > elong | Structured version Visualization version GIF version |
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
elong | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordeq 6378 | . 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴)) | |
2 | df-on 6375 | . 2 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
3 | 1, 2 | elab2g 3666 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 Ord word 6370 Oncon0 6371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-v 3463 df-ss 3961 df-uni 4910 df-tr 5267 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 df-on 6375 |
This theorem is referenced by: elon 6380 eloni 6381 elon2 6382 ordelon 6395 onin 6402 limelon 6435 ordsssuc2 6462 onprc 7781 ssonuni 7783 sucexeloni 7813 sucexeloniOLD 7814 suceloniOLD 7816 ordsucOLD 7818 cofon1 8693 cofon2 8694 enp1i 9304 oion 9561 hartogs 9569 card2on 9579 tskwe 9975 onssnum 10065 hsmexlem1 10451 ondomon 10588 1stcrestlem 23400 nosupno 27682 noinfno 27697 hfninf 35913 rn1st 44788 |
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