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Mirrors > Home > MPE Home > Th. List > phplem3OLD | Structured version Visualization version GIF version |
Description: Obsolete version of phplem1 9242 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 6453 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | phplem2OLD.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | phplem2OLD.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | phplem2OLD 9253 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
5 | 2 | enref 9024 | . . . 4 ⊢ 𝐴 ≈ 𝐴 |
6 | nnord 7895 | . . . . . 6 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | orddif 6482 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | sneq 4641 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
10 | 9 | difeq2d 4136 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
11 | 10 | eqcoms 2743 | . . . . 5 ⊢ (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
12 | 8, 11 | sylan9eq 2795 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵})) |
13 | 5, 12 | breqtrid 5185 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
14 | 4, 13 | jaodan 959 | . 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 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 {csn 4631 class class class wbr 5148 Ord word 6385 suc csuc 6388 ωcom 7887 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-om 7888 df-en 8985 |
This theorem is referenced by: phplem4OLD 9255 phpOLD 9257 |
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