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Mirrors > Home > MPE Home > Th. List > nnon | Structured version Visualization version GIF version |
Description: A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
nnon | ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 6961 | . 2 ⊢ ω ⊆ On | |
2 | 1 | sseli 3564 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Oncon0 5640 ωcom 6957 |
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-sep 4709 ax-nul 4717 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-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: nnoni 6964 nnord 6965 peano4 6980 findsg 6985 onasuc 7495 onmsuc 7496 nna0 7571 nnm0 7572 nnasuc 7573 nnmsuc 7574 nnesuc 7575 nnecl 7580 nnawordi 7588 nnmword 7600 nnawordex 7604 nnaordex 7605 oaabslem 7610 oaabs 7611 oaabs2 7612 omabslem 7613 omabs 7614 nnneo 7618 nneob 7619 onfin2 8037 findcard3 8088 dffi3 8220 card2inf 8343 elom3 8428 cantnfp1lem3 8460 cnfcomlem 8479 cnfcom 8480 cnfcom3 8484 finnum 8657 cardnn 8672 nnsdomel 8699 nnacda 8906 ficardun2 8908 ackbij1lem15 8939 ackbij2lem2 8945 ackbij2lem3 8946 ackbij2 8948 fin23lem22 9032 isf32lem5 9062 fin1a2lem4 9108 fin1a2lem9 9113 pwfseqlem3 9361 winainflem 9394 wunr1om 9420 tskr1om 9468 grothomex 9530 pion 9580 om2uzlt2i 12612 bnj168 30052 elhf2 31452 findreccl 31622 rdgeqoa 32394 finxpreclem4 32407 finxpreclem6 32409 harinf 36619 |
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