Theorem unidifsnel 32559

Description: The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)

Assertion

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

Proof of Theorem unidifsnel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2onn 8568	. . . . . . . . . 10 ⊢ 2o ∈ ω
2		nnfi 9090	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
4		enfi 9109	. . . . . . . . 9 ⊢ (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
5	3, 4	mpbiri 258	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ Fin)
6	5	adantl 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7		diffi 9097	. . . . . . 7 ⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
9	8	cardidd 10457	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
10	9	ensymd 8940	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11		simpl 482	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
12		dif1card 9918	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
13	6, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14		cardennn 9893	. . . . . . . . 9 ⊢ ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
15	1, 14	mpan2 691	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16		df-2o 8396	. . . . . . . 8 ⊢ 2o = suc 1o
17	15, 16	eqtrdi 2785	. . . . . . 7 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
19	13, 18	eqtr3d 2771	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20		suc11reg 9526	. . . . 5 ⊢ (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
21	19, 20	sylib 218	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
22	10, 21	breqtrd 5122	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23		en1 8959	. . 3 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
24	22, 23	sylib 218	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25		simpr 484	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
26	25	unieqd 4874	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥})
27		unisnv 4881	. . . 4 ⊢ ∪ {𝑥} = 𝑥
28	26, 27	eqtrdi 2785	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥)
29		difssd 4087	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ⊆ 𝑃)
30	25, 29	eqsstrrd 3967	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → {𝑥} ⊆ 𝑃)
31		vsnid 4618	. . . 4 ⊢ 𝑥 ∈ {𝑥}
32		ssel2 3926	. . . 4 ⊢ (({𝑥} ⊆ 𝑃 ∧ 𝑥 ∈ {𝑥}) → 𝑥 ∈ 𝑃)
33	30, 31, 32	sylancl 586	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑥 ∈ 𝑃)
34	28, 33	eqeltrd 2834	. 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
35	24, 34	exlimddv 1936	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  ∪ cuni 4861   class class class wbr 5096  suc csuc 6317  ‘cfv 6490  ωcom 7806  1_oc1o 8388  2_oc2o 8389   ≈ cen 8878  Fincfn 8881  cardccrd 9845

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-ac2 10371

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-ac 10024

This theorem is referenced by:  cyc3genpmlem  33182