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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 4265 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ On 𝜑) | |
2 | ssrab2 3983 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
3 | onint 7373 | . . . 4 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) | |
4 | 2, 3 | mpan 686 | . . 3 ⊢ ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
5 | 1, 4 | sylbir 236 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑}) |
6 | nfrab1 3346 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
7 | 6 | nfint 4798 | . . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝜑} |
8 | nfcv 2951 | . . . 4 ⊢ Ⅎ𝑥On | |
9 | onminsb.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
10 | onminsb.2 | . . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) | |
11 | 7, 8, 9, 10 | elrabf 3617 | . . 3 ⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ On ∧ 𝜓)) |
12 | 11 | simprbi 497 | . 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 207 = wceq 1525 Ⅎwnf 1769 ∈ wcel 2083 ≠ wne 2986 ∃wrex 3108 {crab 3111 ⊆ wss 3865 ∅c0 4217 ∩ cint 4788 Oncon0 6073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-br 4969 df-opab 5031 df-tr 5071 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-ord 6076 df-on 6077 |
This theorem is referenced by: oawordeulem 8037 rankidb 9082 cardmin2 9280 cardaleph 9368 cardmin 9839 |
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