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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 6391 | . 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴)) | |
| 2 | df-on 6388 | . 2 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
| 3 | 1, 2 | elab2g 3680 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Ord word 6383 Oncon0 6384 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: elon 6393 eloni 6394 elon2 6395 ordelon 6408 onin 6415 limelon 6448 ordsssuc2 6475 onprc 7798 ssonuni 7800 sucexeloni 7829 sucexeloniOLD 7830 suceloniOLD 7832 ordsucOLD 7834 cofon1 8710 cofon2 8711 enp1i 9313 oion 9576 hartogs 9584 card2on 9594 tskwe 9990 onssnum 10080 hsmexlem1 10466 ondomon 10603 1stcrestlem 23460 nosupno 27748 noinfno 27763 hfninf 36187 rn1st 45280 |
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