Theorem unidifsnel 32563

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 8698	. . . . . . . . . 10 ⊢ 2o ∈ ω
2		nnfi 9233	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
4		enfi 9253	. . . . . . . . 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 9242	. . . . . . 7 ⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
9	8	cardidd 10618	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
10	9	ensymd 9065	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11		simpl 482	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
12		dif1card 10079	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
13	6, 11, 12	syl2anc 583	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14		cardennn 10052	. . . . . . . . 9 ⊢ ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
15	1, 14	mpan2 690	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16		df-2o 8523	. . . . . . . 8 ⊢ 2o = suc 1o
17	15, 16	eqtrdi 2796	. . . . . . 7 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
19	13, 18	eqtr3d 2782	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20		suc11reg 9688	. . . . 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 5192	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23		en1 9086	. . 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 4944	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥})
27		unisnv 4951	. . . 4 ⊢ ∪ {𝑥} = 𝑥
28	26, 27	eqtrdi 2796	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥)
29		difssd 4160	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ⊆ 𝑃)
30	25, 29	eqsstrrd 4048	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → {𝑥} ⊆ 𝑃)
31		vsnid 4685	. . . 4 ⊢ 𝑥 ∈ {𝑥}
32		ssel2 4003	. . . 4 ⊢ (({𝑥} ⊆ 𝑃 ∧ 𝑥 ∈ {𝑥}) → 𝑥 ∈ 𝑃)
33	30, 31, 32	sylancl 585	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑥 ∈ 𝑃)
34	28, 33	eqeltrd 2844	. 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
35	24, 34	exlimddv 1934	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ∪ cuni 4931   class class class wbr 5166  suc csuc 6397  ‘cfv 6573  ωcom 7903  1_oc1o 8515  2_oc2o 8516   ≈ cen 9000  Fincfn 9003  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-ac 10185

This theorem is referenced by:  cyc3genpmlem  33144