Theorem en2eleq 9434

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 8266	. . . . . 6 ⊢ 2o ∈ ω
2		nnfi 8711	. . . . . 6 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2o ∈ Fin
4		enfi 8734	. . . . 5 ⊢ (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
5	3, 4	mpbiri 260	. . . 4 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ Fin)
6	5	adantl 484	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7		simpl 485	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
8		1onn 8265	. . . . . . . 8 ⊢ 1o ∈ ω
9		simpr 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ 2o)
10		df-2o 8103	. . . . . . . . 9 ⊢ 2o = suc 1o
11	9, 10	breqtrdi 5107	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ≈ suc 1o)
12		dif1en 8751	. . . . . . . 8 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1o)
13	8, 11, 7, 12	mp3an2i 1462	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
14		en1uniel 8581	. . . . . . 7 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
16		eldifsn 4719	. . . . . 6 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) ↔ (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
17	15, 16	sylib 220	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋))
18	17	simpld 497	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
19	7, 18	prssd 4755	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃)
20	17	simprd 498	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
21	20	necomd 3071	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
22		pr2nelem 9430	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o)
23	7, 18, 21, 22	syl3anc 1367	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o)
24		ensym 8558	. . . . 5 ⊢ (𝑃 ≈ 2o → 2o ≈ 𝑃)
25	24	adantl 484	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 2o ≈ 𝑃)
26		entr 8561	. . . 4 ⊢ (({𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 2o ∧ 2o ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
27	23, 25, 26	syl2anc 586	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃)
28		fisseneq 8729	. . 3 ⊢ ((𝑃 ∈ Fin ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ⊆ 𝑃 ∧ {𝑋, ∪ (𝑃 ∖ {𝑋})} ≈ 𝑃) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
29	6, 19, 27, 28	syl3anc 1367	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → {𝑋, ∪ (𝑃 ∖ {𝑋})} = 𝑃)
30	29	eqcomd 2827	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∖ cdif 3933   ⊆ wss 3936  {csn 4567  {cpr 4569  ∪ cuni 4838   class class class wbr 5066  suc csuc 6193  ωcom 7580  1_oc1o 8095  2_oc2o 8096   ≈ cen 8506  Fincfn 8509

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-1o 8102  df-2o 8103  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513

This theorem is referenced by:  en2other2  9435  psgnunilem1  18621  cyc3genpmlem  30793