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Mirrors > Home > MPE Home > Th. List > abf | Structured version Visualization version GIF version |
Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2112, ax-10 2141, ax-11 2158, ax-12 2175. (Revised by Gino Giotto, 30-Jun-2024.) |
Ref | Expression |
---|---|
abf.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
abf | ⊢ {𝑥 ∣ 𝜑} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abf.1 | . . . 4 ⊢ ¬ 𝜑 | |
2 | 1 | bifal 1559 | . . 3 ⊢ (𝜑 ↔ ⊥) |
3 | 2 | abbii 2808 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥} |
4 | dfnul4 4236 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
5 | 3, 4 | eqtr4i 2768 | 1 ⊢ {𝑥 ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ⊥wfal 1555 {cab 2714 ∅c0 4234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-dif 3866 df-nul 4235 |
This theorem is referenced by: csbprc 4318 mpo0 7293 fi0 9033 join0 17908 meet0 17909 fmla0disjsuc 33070 addsid1 33861 0qs 36235 pmapglb2xN 37521 |
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