Theorem unidifsnne 30296

Description: The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)

Assertion

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

Proof of Theorem unidifsnne

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2onn 8266	. . . . . . . . . 10 ⊢ 2o ∈ ω
2		nnfi 8711	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
4		enfi 8734	. . . . . . . . 9 ⊢ (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
5	3, 4	mpbiri 260	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ Fin)
6	5	adantl 484	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7		diffi 8750	. . . . . . 7 ⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
9	8	cardidd 9971	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
10	9	ensymd 8560	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11		simpl 485	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
12		dif1card 9436	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
13	6, 11, 12	syl2anc 586	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14		cardennn 9412	. . . . . . . . 9 ⊢ ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
15	1, 14	mpan2 689	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16		df-2o 8103	. . . . . . . 8 ⊢ 2o = suc 1o
17	15, 16	syl6eq 2872	. . . . . . 7 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
18	17	adantl 484	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
19	13, 18	eqtr3d 2858	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20		suc11reg 9082	. . . . 5 ⊢ (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
21	19, 20	sylib 220	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
22	10, 21	breqtrd 5092	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
23		en1 8576	. . 3 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
24	22, 23	sylib 220	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
25		simplll 773	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ 𝑃)
26	25	elexd 3514	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → 𝑋 ∈ V)
27		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑥})
28		sneqbg 4774	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑃 → ({𝑋} = {𝑥} ↔ 𝑋 = 𝑥))
29	28	biimpar 480	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
30	29	ad4ant14 750	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = {𝑥})
31	27, 30	eqtr4d 2859	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → (𝑃 ∖ {𝑋}) = {𝑋})
32	31	ineq2d 4189	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ({𝑋} ∩ {𝑋}))
33		disjdif 4421	. . . . . . . . 9 ⊢ ({𝑋} ∩ (𝑃 ∖ {𝑋})) = ∅
34		inidm 4195	. . . . . . . . 9 ⊢ ({𝑋} ∩ {𝑋}) = {𝑋}
35	32, 33, 34	3eqtr3g 2879	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ∅ = {𝑋})
36	35	eqcomd 2827	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → {𝑋} = ∅)
37		snprc 4653	. . . . . . 7 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅)
38	36, 37	sylibr 236	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) ∧ 𝑋 = 𝑥) → ¬ 𝑋 ∈ V)
39	26, 38	pm2.65da 815	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ¬ 𝑋 = 𝑥)
40	39	neqned 3023	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 ≠ 𝑥)
41		simpr 487	. . . . . 6 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
42	41	unieqd 4852	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥})
43		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
44	43	unisn 4858	. . . . 5 ⊢ ∪ {𝑥} = 𝑥
45	42, 44	syl6eq 2872	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥)
46	40, 45	neeqtrrd 3090	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
47	46	necomd 3071	. 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
48	24, 47	exlimddv 1936	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  Vcvv 3494   ∖ cdif 3933   ∩ cin 3935  ∅c0 4291  {csn 4567  ∪ cuni 4838   class class class wbr 5066  suc csuc 6193  ‘cfv 6355  ωcom 7580  1_oc1o 8095  2_oc2o 8096   ≈ cen 8506  Fincfn 8509  cardccrd 9364

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-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-ac2 9885

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-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  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-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  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-pred 6148  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-isom 6364  df-riota 7114  df-om 7581  df-wrecs 7947  df-recs 8008  df-1o 8102  df-2o 8103  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-ac 9542

This theorem is referenced by:  cyc3genpmlem  30793