|   | Mathbox for Steven Nguyen | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-tz6.12-2 | Structured version Visualization version GIF version | ||
| Description: tz6.12-2 6894 without ax-10 2141, ax-11 2157, ax-12 2177. Improves 118 theorems. (Contributed by SN, 27-May-2025.) | 
| Ref | Expression | 
|---|---|
| sn-tz6.12-2 | ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦)) | |
| 2 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑧)) | |
| 3 | 1, 2 | euabsn2w 42689 | . . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦}) | 
| 4 | 3 | notbii 320 | . 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦}) | 
| 5 | df-fv 6569 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
| 6 | iotanul2 6531 | . . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (℩𝑥𝐴𝐹𝑥) = ∅) | |
| 7 | 5, 6 | eqtrid 2789 | . 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (𝐹‘𝐴) = ∅) | 
| 8 | 4, 7 | sylbi 217 | 1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∃wex 1779 ∃!weu 2568 {cab 2714 ∅c0 4333 {csn 4626 class class class wbr 5143 ℩cio 6512 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |