Theorem en2eleq 9899

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 8557	. . . . . 6 ⊢ 2o ∈ ω
2		nnfi 9077	. . . . . 6 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2o ∈ Fin
4		enfi 9096	. . . . 5 ⊢ (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
5	3, 4	mpbiri 258	. . . 4 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ Fin)
6	5	adantl 481	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7		simpl 482	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
8		1onn 8555	. . . . . . . 8 ⊢ 1o ∈ ω
9		simpr 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
10		df-2o 8386	. . . . . . . . 9 ⊢ 2o = suc 1o
11	9, 10	breqtrdi 5132	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ suc 1o)
12		dif1ennn 9072	. . . . . . . 8 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1o)
13	8, 11, 7, 12	mp3an2i 1468	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
14		en1uniel 8951	. . . . . . 7 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
16		eldifsn 4738	. . . . . 6 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
17	15, 16	sylib 218	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
18	17	simpld 494	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
19	7, 18	prssd 4774	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃)
20	17	simprd 495	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
21	20	necomd 2983	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
22		enpr2 9895	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o)
23	7, 18, 21, 22	syl3anc 1373	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o)
24		ensym 8925	. . . . 5 ⊢ (𝑃 ≈ 2o → 2o ≈ 𝑃)
25	24	adantl 481	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 2o ≈ 𝑃)
26		entr 8928	. . . 4 ⊢ (({𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o ∧ 2o ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
27	23, 25, 26	syl2anc 584	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
28		fisseneq 9147	. . 3 ⊢ ((𝑃 ∈ Fin ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
29	6, 19, 27, 28	syl3anc 1373	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
30	29	eqcomd 2737	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   ∖ cdif 3899   ⊆ wss 3902  {csn 4576  {cpr 4578  ∪ cuni 4859   class class class wbr 5091  suc csuc 6308  ωcom 7796  1_oc1o 8378  2_oc2o 8379   ≈ cen 8866  Fincfn 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-2o 8386  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873

This theorem is referenced by:  en2other2  9900  psgnunilem1  19406  cyc3genpmlem  33118