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Mirrors > Home > ILE Home > Th. List > omex | GIF version |
Description: The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Ref | Expression |
---|---|
omex | ⊢ ω ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfinf2 4498 | . . 3 ⊢ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) | |
2 | intexabim 4072 | . . 3 ⊢ (∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ∈ V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ∈ V |
4 | dfom3 4501 | . . 3 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} | |
5 | 4 | eleq1i 2203 | . 2 ⊢ (ω ∈ V ↔ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ∈ V) |
6 | 3, 5 | mpbir 145 | 1 ⊢ ω ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∃wex 1468 ∈ wcel 1480 {cab 2123 ∀wral 2414 Vcvv 2681 ∅c0 3358 ∩ cint 3766 suc csuc 4282 ωcom 4499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-int 3767 df-iom 4500 |
This theorem is referenced by: peano5 4507 omelon 4517 frecex 6284 frecabex 6288 fict 6755 infnfi 6782 ominf 6783 inffiexmid 6793 omp1eom 6973 difinfsn 6978 0ct 6985 ctmlemr 6986 ctssdclemn0 6988 ctssdclemr 6990 ctssdc 6991 enumct 6993 omct 6995 ctfoex 6996 exmidlpo 7008 infnninf 7015 nnnninf 7016 niex 7113 enq0ex 7240 nq0ex 7241 uzenom 10191 frecfzennn 10192 nnenom 10200 fxnn0nninf 10204 0tonninf 10205 1tonninf 10206 inftonninf 10207 hashinfuni 10516 hashinfom 10517 xpct 11898 ennnfonelemj0 11903 ennnfonelemg 11905 ennnfonelemen 11923 ctiunct 11942 subctctexmid 13185 0nninf 13186 nnsf 13188 peano4nninf 13189 peano3nninf 13190 nninfex 13194 nninfself 13198 nninfsellemeq 13199 nninfsellemeqinf 13201 sbthom 13210 |
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