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Mirrors > Home > MPE Home > Th. List > dfom5 | Structured version Visualization version GIF version |
Description: ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
dfom5 | ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elom3 9095 | . . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)) | |
2 | vex 3444 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 2 | elintab 4849 | . . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)) |
4 | 1, 3 | bitr4i 281 | . 2 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥}) |
5 | 4 | eqriv 2795 | 1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 ∩ cint 4838 Lim wlim 6160 ωcom 7560 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 |
This theorem is referenced by: (None) |
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