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Mirrors > Home > MPE Home > Th. List > onintrab | Structured version Visualization version GIF version |
Description: The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.) |
Ref | Expression |
---|---|
onintrab | ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intex 5232 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ V) | |
2 | ssrab2 4055 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
3 | oninton 7509 | . . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
5 | 1, 4 | sylbir 237 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
6 | elex 3512 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ V) | |
7 | 5, 6 | impbii 211 | 1 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 ∩ cint 4868 Oncon0 6185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 |
This theorem is referenced by: onintrab2 7511 sltval2 33158 sltres 33164 |
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