Theorem pr2cv1 7391

Description: If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)

Assertion

Ref	Expression
pr2cv1	⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V)

Proof of Theorem pr2cv1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2o3 6592	. . . 4 ⊢ 2o = {∅, 1o}
2		ensym 6950	. . . 4 ⊢ ({𝐴, 𝐵} ≈ 2o → 2o ≈ {𝐴, 𝐵})
3	1, 2	eqbrtrrid 4122	. . 3 ⊢ ({𝐴, 𝐵} ≈ 2o → {∅, 1o} ≈ {𝐴, 𝐵})
4		bren 6912	. . 3 ⊢ ({∅, 1o} ≈ {𝐴, 𝐵} ↔ ∃𝑓 𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵})
5	3, 4	sylib 122	. 2 ⊢ ({𝐴, 𝐵} ≈ 2o → ∃𝑓 𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵})
6		vex 2803	. . . . . . 7 ⊢ 𝑓 ∈ V
7		0ex 4214	. . . . . . 7 ⊢ ∅ ∈ V
8	6, 7	fvex 5655	. . . . . 6 ⊢ (𝑓‘∅) ∈ V
9		eleq1 2292	. . . . . 6 ⊢ ((𝑓‘∅) = 𝐴 → ((𝑓‘∅) ∈ V ↔ 𝐴 ∈ V))
10	8, 9	mpbii 148	. . . . 5 ⊢ ((𝑓‘∅) = 𝐴 → 𝐴 ∈ V)
11	10	adantl 277	. . . 4 ⊢ ((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐴) → 𝐴 ∈ V)
12		1oex 6585	. . . . . . . 8 ⊢ 1o ∈ V
13	6, 12	fvex 5655	. . . . . . 7 ⊢ (𝑓‘1o) ∈ V
14		eleq1 2292	. . . . . . 7 ⊢ ((𝑓‘1o) = 𝐴 → ((𝑓‘1o) ∈ V ↔ 𝐴 ∈ V))
15	13, 14	mpbii 148	. . . . . 6 ⊢ ((𝑓‘1o) = 𝐴 → 𝐴 ∈ V)
16	15	adantl 277	. . . . 5 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐴) → 𝐴 ∈ V)
17		simplr 528	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘∅) = 𝐵)
18		simpr 110	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘1o) = 𝐵)
19	17, 18	eqtr4d 2265	. . . . . . 7 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘∅) = (𝑓‘1o))
20		f1of1 5579	. . . . . . . . 9 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝑓:{∅, 1o}–1-1→{𝐴, 𝐵})
21	20	ad2antrr 488	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝑓:{∅, 1o}–1-1→{𝐴, 𝐵})
22	7	prid1 3775	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
23	22	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ∅ ∈ {∅, 1o})
24	12	prid2 3776	. . . . . . . . 9 ⊢ 1o ∈ {∅, 1o}
25	24	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 1o ∈ {∅, 1o})
26		f1veqaeq 5905	. . . . . . . 8 ⊢ ((𝑓:{∅, 1o}–1-1→{𝐴, 𝐵} ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})) → ((𝑓‘∅) = (𝑓‘1o) → ∅ = 1o))
27	21, 23, 25, 26	syl12anc 1269	. . . . . . 7 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ((𝑓‘∅) = (𝑓‘1o) → ∅ = 1o))
28	19, 27	mpd 13	. . . . . 6 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ∅ = 1o)
29		1n0 6595	. . . . . . . 8 ⊢ 1o ≠ ∅
30	29	nesymi 2446	. . . . . . 7 ⊢ ¬ ∅ = 1o
31	30	a1i 9	. . . . . 6 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ¬ ∅ = 1o)
32	28, 31	pm2.21dd 623	. . . . 5 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝐴 ∈ V)
33		f1of 5580	. . . . . . . 8 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝑓:{∅, 1o}⟶{𝐴, 𝐵})
34	24	a1i 9	. . . . . . . 8 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 1o ∈ {∅, 1o})
35	33, 34	ffvelcdmd 5779	. . . . . . 7 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘1o) ∈ {𝐴, 𝐵})
36		elpri 3690	. . . . . . 7 ⊢ ((𝑓‘1o) ∈ {𝐴, 𝐵} → ((𝑓‘1o) = 𝐴 ∨ (𝑓‘1o) = 𝐵))
37	35, 36	syl 14	. . . . . 6 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → ((𝑓‘1o) = 𝐴 ∨ (𝑓‘1o) = 𝐵))
38	37	adantr 276	. . . . 5 ⊢ ((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) → ((𝑓‘1o) = 𝐴 ∨ (𝑓‘1o) = 𝐵))
39	16, 32, 38	mpjaodan 803	. . . 4 ⊢ ((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) → 𝐴 ∈ V)
40	22	a1i 9	. . . . . 6 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → ∅ ∈ {∅, 1o})
41	33, 40	ffvelcdmd 5779	. . . . 5 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘∅) ∈ {𝐴, 𝐵})
42		elpri 3690	. . . . 5 ⊢ ((𝑓‘∅) ∈ {𝐴, 𝐵} → ((𝑓‘∅) = 𝐴 ∨ (𝑓‘∅) = 𝐵))
43	41, 42	syl 14	. . . 4 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → ((𝑓‘∅) = 𝐴 ∨ (𝑓‘∅) = 𝐵))
44	11, 39, 43	mpjaodan 803	. . 3 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝐴 ∈ V)
45	44	exlimiv 1644	. 2 ⊢ (∃𝑓 𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝐴 ∈ V)
46	5, 45	syl 14	1 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800  ∅c0 3492  {cpr 3668   class class class wbr 4086  –1-1→wf1 5321  –1-1-onto→wf1o 5323  ‘cfv 5324  1_oc1o 6570  2_oc2o 6571   ≈ cen 6902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905

This theorem is referenced by:  pr2cv2  7392  pr2cv  7393