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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 4395 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑) | |
2 | ssrab2 4090 | . . . 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 3454 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
7 | 6 | nfint 4961 | . . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑} |
8 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥On | |
9 | onminsb.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
10 | onminsb.2 | . . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | |
11 | 7, 8, 9, 10 | elrabf 3691 | . . 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 1537 Ⅎwnf 1780 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 ⊆ wss 3963 ∅c0 4339 ∩ cint 4951 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: oawordeulem 8591 rankidb 9838 cardmin2 10037 cardaleph 10127 cardmin 10602 naddwordnexlem4 43391 |
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