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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 5222 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ V) | |
2 | onintrab 7569 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
3 | 1, 2 | bitri 278 | 1 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2110 ∃wrex 3055 {crab 3058 Vcvv 3401 ∩ cint 4849 Oncon0 6202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-tr 5151 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-ord 6205 df-on 6206 |
This theorem is referenced by: oeeulem 8318 cardmin2 9598 cardaleph 9686 cardmin 10161 naddcllem 33525 nosepon 33562 |
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