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Mirrors > Home > MPE Home > Th. List > phplem1OLD | Structured version Visualization version GIF version |
Description: Obsolete lemma for php 9212 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 7865 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | nordeq 6382 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | |
3 | disjsn2 4715 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
5 | 1, 4 | sylan 578 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
6 | undif4 4465 | . . 3 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵})) | |
7 | df-suc 6369 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
8 | 7 | equncomi 4154 | . . . 4 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) |
9 | 8 | difeq1i 4117 | . . 3 ⊢ (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵}) |
10 | 6, 9 | eqtr4di 2788 | . 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 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 Ord word 6362 suc csuc 6365 ωcom 7857 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6366 df-on 6367 df-suc 6369 df-om 7858 |
This theorem is referenced by: phplem2OLD 9220 |
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