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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 6191 | . 2 ⊢ (𝑥 = 𝐴 → (Ord 𝑥 ↔ Ord 𝐴)) | |
2 | df-on 6188 | . 2 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
3 | 1, 2 | elab2g 3665 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Ord word 6183 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-in 3940 df-ss 3949 df-uni 4831 df-tr 5164 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: elon 6193 eloni 6194 elon2 6195 ordelon 6208 onin 6215 limelon 6247 ordsssuc2 6272 onprc 7488 ssonuni 7490 suceloni 7517 ordsuc 7518 oion 8988 hartogs 8996 card2on 9006 tskwe 9367 onssnum 9454 hsmexlem1 9836 ondomon 9973 1stcrestlem 21988 nosupno 33100 hfninf 33544 |
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