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Mirrors > Home > MPE Home > Th. List > onintrab2 | Structured version Visualization version GIF version |
Description: An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.) |
Ref | Expression |
---|---|
onintrab2 | ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intexrab 5353 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ V) | |
2 | onintrab 7816 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 ∃wrex 3068 {crab 3433 Vcvv 3478 ∩ cint 4951 Oncon0 6386 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: oeeulem 8638 cofon1 8709 cofon2 8710 cofonr 8711 naddcllem 8713 naddunif 8730 cardmin2 10037 cardaleph 10127 cardmin 10602 nosepon 27725 minregex 43524 |
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