Theorem unidifsnel 32514

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 8583	. . . . . . . . . 10 ⊢ 2o ∈ ω
2		nnfi 9108	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
4		enfi 9128	. . . . . . . . 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 9116	. . . . . . 7 ⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
9	8	cardidd 10478	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
10	9	ensymd 8953	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11		simpl 482	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
12		dif1card 9939	. . . . . . 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 9912	. . . . . . . . 9 ⊢ ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
15	1, 14	mpan2 691	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16		df-2o 8412	. . . . . . . 8 ⊢ 2o = suc 1o
17	15, 16	eqtrdi 2780	. . . . . . 7 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
19	13, 18	eqtr3d 2766	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20		suc11reg 9548	. . . . 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 5128	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23		en1 8972	. . 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 4880	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥})
27		unisnv 4887	. . . 4 ⊢ ∪ {𝑥} = 𝑥
28	26, 27	eqtrdi 2780	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥)
29		difssd 4096	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ⊆ 𝑃)
30	25, 29	eqsstrrd 3979	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → {𝑥} ⊆ 𝑃)
31		vsnid 4623	. . . 4 ⊢ 𝑥 ∈ {𝑥}
32		ssel2 3938	. . . 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 2828	. 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
35	24, 34	exlimddv 1935	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∖ cdif 3908   ⊆ wss 3911  {csn 4585  ∪ cuni 4867   class class class wbr 5102  suc csuc 6322  ‘cfv 6499  ωcom 7822  1_oc1o 8404  2_oc2o 8405   ≈ cen 8892  Fincfn 8895  cardccrd 9864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-ac2 10392

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-ac 10045

This theorem is referenced by:  cyc3genpmlem  33123