Theorem unidifsnel 33229

Description: The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.) Remove dependency on ax-reg 9623. (Revised by Timothy J. O'Malley, 15-Oct-2026.)

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 8689	. . . . . . . . . 10 ⊢ 2o ∈ ω
2		nnfi 9221	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
4		enfi 9240	. . . . . . . . 9 ⊢ (𝑃 ≈ 2o → (𝑃 ∈ Fin ↔ 2o ∈ Fin))
5	3, 4	mpbiri 261	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ Fin)
6	5	adantl 487	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 ∈ Fin)
7		diffi 9228	. . . . . . 7 ⊢ (𝑃 ∈ Fin → (𝑃 ∖ {𝑋}) ∈ Fin)
8	6, 7	syl 18	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ Fin)
9	8	cardidd 10672	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) ≈ (𝑃 ∖ {𝑋}))
10	9	ensymd 9070	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ (card‘(𝑃 ∖ {𝑋})))
11		simpl 488	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑋 ∈ 𝑃)
12		dif1card 10128	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ 𝑋 ∈ 𝑃) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
13	6, 11, 12	syl2anc 596	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc (card‘(𝑃 ∖ {𝑋})))
14		cardennn 10103	. . . . . . . . 9 ⊢ ((𝑃 ≈ 2o ∧ 2o ∈ ω) → (card‘𝑃) = 2o)
15	1, 14	mpan2 704	. . . . . . . 8 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = 2o)
16		df-2o 8515	. . . . . . . 8 ⊢ 2o = suc 1o
17	15, 16	eqtrdi 2814	. . . . . . 7 ⊢ (𝑃 ≈ 2o → (card‘𝑃) = suc 1o)
18	17	adantl 487	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘𝑃) = suc 1o)
19	13, 18	eqtr3d 2800	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → suc (card‘(𝑃 ∖ {𝑋})) = suc 1o)
20		cardon 10062	. . . . . 6 ⊢ (card‘(𝑃 ∖ {𝑋})) ∈ On
21		1on 8527	. . . . . 6 ⊢ 1o ∈ On
22		suc11 6498	. . . . . 6 ⊢ (((card‘(𝑃 ∖ {𝑋})) ∈ On ∧ 1o ∈ On) → (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o))
23	20, 21, 22	mp2an 705	. . . . 5 ⊢ (suc (card‘(𝑃 ∖ {𝑋})) = suc 1o ↔ (card‘(𝑃 ∖ {𝑋})) = 1o)
24	19, 23	sylib 221	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (card‘(𝑃 ∖ {𝑋})) = 1o)
25	10, 24	breqtrd 5137	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≈ 1o)
26		en1 9089	. . 3 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1o ↔ ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
27	25, 26	sylib 221	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∃𝑥(𝑃 ∖ {𝑋}) = {𝑥})
28		simpr 490	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) = {𝑥})
29	28	unieqd 4885	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = ∪ {𝑥})
30		unisnv 4892	. . . 4 ⊢ ∪ {𝑥} = 𝑥
31	29, 30	eqtrdi 2814	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) = 𝑥)
32		difssd 4076	. . . . 5 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → (𝑃 ∖ {𝑋}) ⊆ 𝑃)
33	28, 32	eqsstrrd 3948	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → {𝑥} ⊆ 𝑃)
34		vsnid 4625	. . . 4 ⊢ 𝑥 ∈ {𝑥}
35		ssel2 3908	. . . 4 ⊢ (({𝑥} ⊆ 𝑃 ∧ 𝑥 ∈ {𝑥}) → 𝑥 ∈ 𝑃)
36	33, 34, 35	sylancl 598	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → 𝑥 ∈ 𝑃)
37	31, 36	eqeltrd 2864	. 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) ∧ (𝑃 ∖ {𝑋}) = {𝑥}) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
38	27, 37	exlimddv 1970	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  ∪ cuni 4872   class class class wbr 5108  Oncon0 6388  suc csuc 6390  ‘cfv 6566  ωcom 7911  1_oc1o 8507  2_oc2o 8508   ≈ cen 9008  Fincfn 9011  cardccrd 10053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-ac2 10580

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-ac 10234

This theorem is used by:  cyc3genpmlem  33811