|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > phplem1OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete lemma for php 9247 as of 22-Nov-2024. (Contributed by NM, 25-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| phplem1OLD | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nnord 7895 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
| 2 | nordeq 6403 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | |
| 3 | disjsn2 4712 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) | 
| 5 | 1, 4 | sylan 580 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) | 
| 6 | undif4 4467 | . . 3 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵})) | |
| 7 | df-suc 6390 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 8 | 7 | equncomi 4160 | . . . 4 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) | 
| 9 | 8 | difeq1i 4122 | . . 3 ⊢ (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵}) | 
| 10 | 6, 9 | eqtr4di 2795 | . 2 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) | 
| 11 | 5, 10 | syl 17 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 Ord word 6383 suc csuc 6386 ωcom 7887 | 
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 df-om 7888 | 
| This theorem is referenced by: phplem2OLD 9255 | 
| Copyright terms: Public domain | W3C validator |