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| Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.) | 
| Ref | Expression | 
|---|---|
| onnminsb.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| onnminsb | ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | onnminsb.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | elrab 3691 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓)) | 
| 3 | ssrab2 4079 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
| 4 | onnmin 7819 | . . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) | |
| 5 | 3, 4 | mpan 690 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) | 
| 6 | 2, 5 | sylbir 235 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) | 
| 7 | 6 | ex 412 | . 2 ⊢ (𝐴 ∈ On → (𝜓 → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})) | 
| 8 | 7 | con2d 134 | 1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 ∩ cint 4945 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: onminex 7823 oawordeulem 8593 oeeulem 8640 nnawordex 8676 tcrank 9925 alephnbtwn 10112 cardaleph 10130 cardmin 10605 sltval2 27702 nosepeq 27731 nosupbnd2lem1 27761 noinfbnd2lem1 27776 naddwordnexlem4 43419 | 
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