Theorem en2eleq 9013

Description: Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
en2eleq	⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})

Proof of Theorem en2eleq

Step	Hyp	Ref	Expression
1		2onn 7881	. . . . . 6 ⊢ 2𝑜 ∈ ω
2		nnfi 8310	. . . . . 6 ⊢ (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2𝑜 ∈ Fin
4		enfi 8333	. . . . 5 ⊢ (𝑃 ≈ 2𝑜 → (𝑃 ∈ Fin ↔ 2𝑜 ∈ Fin))
5	3, 4	mpbiri 248	. . . 4 ⊢ (𝑃 ≈ 2𝑜 → 𝑃 ∈ Fin)
6	5	adantl 473	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ∈ Fin)
7		simpl 474	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ 𝑃)
8		1onn 7880	. . . . . . . . 9 ⊢ 1𝑜 ∈ ω
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10		simpr 479	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11		df-2o 7722	. . . . . . . . 9 ⊢ 2𝑜 = suc 1𝑜
12	10, 11	syl6breq 4837	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13		dif1en 8350	. . . . . . . 8 ⊢ ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜 ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
14	9, 12, 7, 13	syl3anc 1473	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15		en1uniel 8185	. . . . . . 7 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1𝑜 → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
17		eldifsn 4454	. . . . . 6 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
18	16, 17	sylib 208	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
19	18	simpld 477	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
20		prssi 4490	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃)
21	7, 19, 20	syl2anc 696	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃)
22	18	simprd 482	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
23	22	necomd 2979	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
24		pr2nelem 9009	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2𝑜)
25	7, 19, 23, 24	syl3anc 1473	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2𝑜)
26		ensym 8162	. . . . 5 ⊢ (𝑃 ≈ 2𝑜 → 2𝑜 ≈ 𝑃)
27	26	adantl 473	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 2𝑜 ≈ 𝑃)
28		entr 8165	. . . 4 ⊢ (({𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2𝑜 ∧ 2𝑜 ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
29	25, 27, 28	syl2anc 696	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
30		fisseneq 8328	. . 3 ⊢ ((𝑃 ∈ Fin ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
31	6, 21, 29, 30	syl3anc 1473	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
32	31	eqcomd 2758	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ≠ wne 2924   ∖ cdif 3704   ⊆ wss 3707  {csn 4313  {cpr 4315  ∪ cuni 4580   class class class wbr 4796  suc csuc 5878  ωcom 7222  1_𝑜c1o 7714  2_𝑜c2o 7715   ≈ cen 8110  Fincfn 8113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117

This theorem is referenced by:  en2other2  9014  psgnunilem1  18105