Theorem pr2cv1 7324

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 6534	. . . 4 ⊢ 2o = {∅, 1o}
2		ensym 6891	. . . 4 ⊢ ({𝐴, 𝐵} ≈ 2o → 2o ≈ {𝐴, 𝐵})
3	1, 2	eqbrtrrid 4090	. . 3 ⊢ ({𝐴, 𝐵} ≈ 2o → {∅, 1o} ≈ {𝐴, 𝐵})
4		bren 6853	. . 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 2776	. . . . . . 7 ⊢ 𝑓 ∈ V
7		0ex 4182	. . . . . . 7 ⊢ ∅ ∈ V
8	6, 7	fvex 5614	. . . . . 6 ⊢ (𝑓‘∅) ∈ V
9		eleq1 2269	. . . . . 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 6528	. . . . . . . 8 ⊢ 1o ∈ V
13	6, 12	fvex 5614	. . . . . . 7 ⊢ (𝑓‘1o) ∈ V
14		eleq1 2269	. . . . . . 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 2242	. . . . . . 7 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘∅) = (𝑓‘1o))
20		f1of1 5538	. . . . . . . . 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 3744	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
23	22	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ∅ ∈ {∅, 1o})
24	12	prid2 3745	. . . . . . . . 9 ⊢ 1o ∈ {∅, 1o}
25	24	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 1o ∈ {∅, 1o})
26		f1veqaeq 5856	. . . . . . . 8 ⊢ ((𝑓:{∅, 1o}–1-1→{𝐴, 𝐵} ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})) → ((𝑓‘∅) = (𝑓‘1o) → ∅ = 1o))
27	21, 23, 25, 26	syl12anc 1248	. . . . . . 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 6536	. . . . . . . 8 ⊢ 1o ≠ ∅
30	29	nesymi 2423	. . . . . . 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 621	. . . . 5 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝐴 ∈ V)
33		f1of 5539	. . . . . . . 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 5734	. . . . . . 7 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘1o) ∈ {𝐴, 𝐵})
36		elpri 3661	. . . . . . 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 800	. . . 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 5734	. . . . 5 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘∅) ∈ {𝐴, 𝐵})
42		elpri 3661	. . . . 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 800	. . 3 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝐴 ∈ V)
45	44	exlimiv 1622	. 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 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  ∅c0 3464  {cpr 3639   class class class wbr 4054  –1-1→wf1 5282  –1-1-onto→wf1o 5284  ‘cfv 5285  1_oc1o 6513  2_oc2o 6514   ≈ cen 6843

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-1o 6520  df-2o 6521  df-er 6638  df-en 6846

This theorem is referenced by:  pr2cv2  7325  pr2cv  7326