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Mirrors > Home > MPE Home > Th. List > 2onn | Structured version Visualization version GIF version |
Description: The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7709, see 2onnALT 8627. (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7709. (Revised by BTernaryTau, 1-Dec-2024.) |
Ref | Expression |
---|---|
2onn | ⊢ 2o ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 8464 | . 2 ⊢ 2o ∈ On | |
2 | 2ellim 8483 | . . 3 ⊢ (Lim 𝑥 → 2o ∈ 𝑥) | |
3 | 2 | ax-gen 1797 | . 2 ⊢ ∀𝑥(Lim 𝑥 → 2o ∈ 𝑥) |
4 | elom 7842 | . 2 ⊢ (2o ∈ ω ↔ (2o ∈ On ∧ ∀𝑥(Lim 𝑥 → 2o ∈ 𝑥))) | |
5 | 1, 3, 4 | mpbir2an 709 | 1 ⊢ 2o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2106 Oncon0 6354 Lim wlim 6355 ωcom 7839 2oc2o 8444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-om 7840 df-1o 8450 df-2o 8451 |
This theorem is referenced by: 3onn 8628 nn2m 8638 nnneo 8639 nneob 8640 omopthlem1 8643 omopthlem2 8644 pwen 9135 en2eqpr 9986 en2eleq 9987 unctb 10184 infdjuabs 10185 ackbij1lem5 10203 sdom2en01 10281 fin56 10372 fin67 10374 fin1a2lem4 10382 alephexp1 10558 pwcfsdom 10562 alephom 10564 canthp1lem2 10632 pwxpndom2 10644 hash3 14350 hash2pr 14414 pr2pwpr 14424 rpnnen 16154 rexpen 16155 xpsfrnel 17492 xpscf 17495 symggen 19304 psgnunilem1 19327 simpgnsgd 19931 znfld 21051 hauspwdom 22936 xpsmet 23819 xpsxms 23974 xpsms 23975 unidifsnel 31701 unidifsnne 31702 sat1el2xp 34265 ex-sategoelelomsuc 34312 ex-sategoelel12 34313 1oequni2o 36117 finxpreclem4 36143 finxp3o 36149 wepwso 41620 frlmpwfi 41675 2omomeqom 41888 oenord1ex 41900 oaomoencom 41902 2finon 42036 har2o 42132 |
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