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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 4385 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑) | |
2 | ssrab2 4077 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
3 | onint 7782 | . . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | |
4 | 2, 3 | mpan 687 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
5 | 1, 4 | sylbir 234 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
6 | nfrab1 3450 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
7 | 6 | nfint 4960 | . . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑} |
8 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑥On | |
9 | onminsb.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
10 | onminsb.2 | . . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | |
11 | 7, 8, 9, 10 | elrabf 3679 | . . 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 205 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 {crab 3431 ⊆ wss 3948 ∅c0 4322 ∩ cint 4950 Oncon0 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 |
This theorem is referenced by: oawordeulem 8560 rankidb 9801 cardmin2 10000 cardaleph 10090 cardmin 10565 naddwordnexlem4 42615 |
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