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| Mirrors > Home > MPE Home > Th. List > onminsb | Structured version Visualization version GIF version | ||
| Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| onminsb.1 | ⊢ Ⅎ𝑥𝜓 | 
| onminsb.2 | ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| onminsb | ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabn0 4389 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑) | |
| 2 | ssrab2 4080 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
| 3 | onint 7810 | . . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | |
| 4 | 2, 3 | mpan 690 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | 
| 5 | 1, 4 | sylbir 235 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | 
| 6 | nfrab1 3457 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
| 7 | 6 | nfint 4956 | . . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑} | 
| 8 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥On | |
| 9 | onminsb.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 10 | onminsb.2 | . . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | |
| 11 | 7, 8, 9, 10 | elrabf 3688 | . . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓)) | 
| 12 | 11 | simprbi 496 | . 2 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜓) | 
| 13 | 5, 12 | syl 17 | 1 ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ∩ cint 4946 Oncon0 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 | 
| This theorem is referenced by: oawordeulem 8592 rankidb 9840 cardmin2 10039 cardaleph 10129 cardmin 10604 naddwordnexlem4 43414 | 
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