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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 2109, ax-10 2138, ax-11 2155, ax-12 2172. (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 1558 | . . 3 ⊢ (𝜑 ↔ ⊥) |
3 | 2 | abbii 2803 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥} |
4 | dfnul4 4285 | . 2 ⊢ ∅ = {𝑥 ∣ ⊥} | |
5 | 3, 4 | eqtr4i 2764 | 1 ⊢ {𝑥 ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ⊥wfal 1554 {cab 2710 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-dif 3914 df-nul 4284 |
This theorem is referenced by: csbprc 4367 mpo0 7443 fi0 9361 join0 18299 meet0 18300 addsrid 27298 muls01 27397 muls02 27398 mulsrid 27399 mulslid 27400 fmla0disjsuc 34049 0qs 36877 pmapglb2xN 38281 |
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