Theorem pr2cv1 7531

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 6692	. . . 4 ⊢ 2o = {∅, 1o}
2		ensym 7058	. . . 4 ⊢ ({𝐴, 𝐵} ≈ 2o → 2o ≈ {𝐴, 𝐵})
3	1, 2	eqbrtrrid 4161	. . 3 ⊢ ({𝐴, 𝐵} ≈ 2o → {∅, 1o} ≈ {𝐴, 𝐵})
4		bren 7020	. . 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 2824	. . . . . . 7 ⊢ 𝑓 ∈ V
7		0ex 4255	. . . . . . 7 ⊢ ∅ ∈ V
8	6, 7	fvex 5710	. . . . . 6 ⊢ (𝑓‘∅) ∈ V
9		eleq1 2301	. . . . . 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 6685	. . . . . . . 8 ⊢ 1o ∈ V
13	6, 12	fvex 5710	. . . . . . 7 ⊢ (𝑓‘1o) ∈ V
14		eleq1 2301	. . . . . . 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 533	. . . . . . . 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 2274	. . . . . . 7 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘∅) = (𝑓‘1o))
20		f1of1 5633	. . . . . . . . 9 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝑓:{∅, 1o}–1-1→{𝐴, 𝐵})
21	20	ad2antrr 492	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝑓:{∅, 1o}–1-1→{𝐴, 𝐵})
22	7	prid1 3813	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
23	22	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ∅ ∈ {∅, 1o})
24	12	prid2 3814	. . . . . . . . 9 ⊢ 1o ∈ {∅, 1o}
25	24	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 1o ∈ {∅, 1o})
26		f1veqaeq 5965	. . . . . . . 8 ⊢ ((𝑓:{∅, 1o}–1-1→{𝐴, 𝐵} ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})) → ((𝑓‘∅) = (𝑓‘1o) → ∅ = 1o))
27	21, 23, 25, 26	syl12anc 1276	. . . . . . 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 6695	. . . . . . . 8 ⊢ 1o ≠ ∅
30	29	nesymi 2466	. . . . . . 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 629	. . . . 5 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝐴 ∈ V)
33		f1of 5634	. . . . . . . 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 5835	. . . . . . 7 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘1o) ∈ {𝐴, 𝐵})
36		elpri 3728	. . . . . . 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 810	. . . 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 5835	. . . . 5 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘∅) ∈ {𝐴, 𝐵})
42		elpri 3728	. . . . 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 810	. . 3 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝐴 ∈ V)
45	44	exlimiv 1651	. 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 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  {cpr 3706   class class class wbr 4125  –1-1→wf1 5369  –1-1-onto→wf1o 5371  ‘cfv 5372  1_oc1o 6670  2_oc2o 6671   ≈ cen 7010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013

This theorem is referenced by:  pr2cv2  7532  pr2cv  7533