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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 3912 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑) | |
2 | ssrab2 3650 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
3 | onint 6887 | . . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | |
4 | 2, 3 | mpan 702 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
5 | 1, 4 | sylbir 224 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
6 | nfrab1 3099 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
7 | 6 | nfint 4421 | . . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑} |
8 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥On | |
9 | onminsb.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
10 | onminsb.2 | . . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | |
11 | 7, 8, 9, 10 | elrabf 3329 | . . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓)) |
12 | 11 | simprbi 479 | . 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 195 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 ∩ cint 4410 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: oawordeulem 7521 rankidb 8546 cardmin2 8707 cardaleph 8795 cardmin 9265 |
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