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| Mirrors > Home > MPE Home > Th. List > phplem3OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of phplem1 9244 as of 4-Nov-2024. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| phplem2OLD.1 | ⊢ 𝐴 ∈ V | 
| phplem2OLD.2 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| phplem3OLD | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elsuci 6451 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
| 2 | phplem2OLD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 3 | phplem2OLD.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 4 | 2, 3 | phplem2OLD 9255 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | 
| 5 | 2 | enref 9025 | . . . 4 ⊢ 𝐴 ≈ 𝐴 | 
| 6 | nnord 7895 | . . . . . 6 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
| 7 | orddif 6480 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | |
| 8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) | 
| 9 | sneq 4636 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
| 10 | 9 | difeq2d 4126 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) | 
| 11 | 10 | eqcoms 2745 | . . . . 5 ⊢ (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) | 
| 12 | 8, 11 | sylan9eq 2797 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵})) | 
| 13 | 5, 12 | breqtrid 5180 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | 
| 14 | 4, 13 | jaodan 960 | . 2 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | 
| 15 | 1, 14 | sylan2 593 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 {csn 4626 class class class wbr 5143 Ord word 6383 suc csuc 6386 ωcom 7887 ≈ cen 8982 | 
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-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-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-en 8986 | 
| This theorem is referenced by: phplem4OLD 9257 phpOLD 9259 | 
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