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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 5134 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ V) | |
2 | onintrab 7372 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
3 | 1, 2 | bitri 276 | 1 ⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2081 ∃wrex 3106 {crab 3109 Vcvv 3437 ∩ cint 4782 Oncon0 6066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-br 4963 df-opab 5025 df-tr 5064 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-ord 6069 df-on 6070 |
This theorem is referenced by: oeeulem 8077 cardmin2 9273 cardaleph 9361 cardmin 9832 nosepon 32781 |
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