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Mirrors > Home > MPE Home > Th. List > epfrc | Structured version Visualization version GIF version |
Description: A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.) |
Ref | Expression |
---|---|
epfrc.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
epfrc | ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epfrc.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | frc 5215 | . 2 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅) |
3 | dfin5 3731 | . . . . 5 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥} | |
4 | epel 5165 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
5 | 4 | rabbii 3335 | . . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = {𝑦 ∈ 𝐵 ∣ 𝑦 ∈ 𝑥} |
6 | 3, 5 | eqtr4i 2796 | . . . 4 ⊢ (𝐵 ∩ 𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} |
7 | 6 | eqeq1i 2776 | . . 3 ⊢ ((𝐵 ∩ 𝑥) = ∅ ↔ {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅) |
8 | 7 | rexbii 3189 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦 E 𝑥} = ∅) |
9 | 2, 8 | sylibr 224 | 1 ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 {crab 3065 Vcvv 3351 ∩ cin 3722 ⊆ wss 3723 ∅c0 4063 class class class wbr 4786 E cep 5161 Fr wfr 5205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-eprel 5162 df-fr 5208 |
This theorem is referenced by: wefrc 5243 onfr 5906 epfrs 8771 |
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