Theorem en2other2 8832

Description: Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
en2other2	⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)

Proof of Theorem en2other2

Step	Hyp	Ref	Expression
1		en2eleq 8831	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
2		prcom 4267	. . . . . . 7 ⊢ {𝑋, ∪ (𝑃 ∖ {𝑋})} = {∪ (𝑃 ∖ {𝑋}), 𝑋}
3	1, 2	syl6eq 2672	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {∪ (𝑃 ∖ {𝑋}), 𝑋})
4	3	difeq1d 3727	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}))
5		difprsnss 4329	. . . . 5 ⊢ ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋}
6	4, 5	syl6eqss 3655	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋})
7		simpl 473	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ 𝑃)
8		1onn 7719	. . . . . . . . . 10 ⊢ 1𝑜 ∈ ω
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10		simpr 477	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11		df-2o 7561	. . . . . . . . . 10 ⊢ 2𝑜 = suc 1𝑜
12	10, 11	syl6breq 4694	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13		dif1en 8193	. . . . . . . . 9 ⊢ ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜 ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
14	9, 12, 7, 13	syl3anc 1326	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15		en1uniel 8028	. . . . . . . 8 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1𝑜 → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
16		eldifsni 4320	. . . . . . . 8 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
18	17	necomd 2849	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
19		eldifsn 4317	. . . . . 6 ⊢ (𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ↔ (𝑋 ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})))
20	7, 18, 19	sylanbrc 698	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}))
21	20	snssd 4340	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋} ⊆ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}))
22	6, 21	eqssd 3620	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = {𝑋})
23	22	unieqd 4446	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ∪ {𝑋})
24		unisng 4452	. . 3 ⊢ (𝑋 ∈ 𝑃 → ∪ {𝑋} = 𝑋)
25	24	adantr 481	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ {𝑋} = 𝑋)
26	23, 25	eqtrd 2656	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571  {csn 4177  {cpr 4179  ∪ cuni 4436   class class class wbr 4653  suc csuc 5725  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  pmtrfinv  17881