Theorem pr2cv1 7399

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 6596	. . . 4 ⊢ 2o = {∅, 1o}
2		ensym 6954	. . . 4 ⊢ ({𝐴, 𝐵} ≈ 2o → 2o ≈ {𝐴, 𝐵})
3	1, 2	eqbrtrrid 4124	. . 3 ⊢ ({𝐴, 𝐵} ≈ 2o → {∅, 1o} ≈ {𝐴, 𝐵})
4		bren 6916	. . 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 2805	. . . . . . 7 ⊢ 𝑓 ∈ V
7		0ex 4216	. . . . . . 7 ⊢ ∅ ∈ V
8	6, 7	fvex 5659	. . . . . 6 ⊢ (𝑓‘∅) ∈ V
9		eleq1 2294	. . . . . 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 6589	. . . . . . . 8 ⊢ 1o ∈ V
13	6, 12	fvex 5659	. . . . . . 7 ⊢ (𝑓‘1o) ∈ V
14		eleq1 2294	. . . . . . 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 529	. . . . . . . 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 2267	. . . . . . 7 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → (𝑓‘∅) = (𝑓‘1o))
20		f1of1 5582	. . . . . . . . 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 3777	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1o}
23	22	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → ∅ ∈ {∅, 1o})
24	12	prid2 3778	. . . . . . . . 9 ⊢ 1o ∈ {∅, 1o}
25	24	a1i 9	. . . . . . . 8 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 1o ∈ {∅, 1o})
26		f1veqaeq 5909	. . . . . . . 8 ⊢ ((𝑓:{∅, 1o}–1-1→{𝐴, 𝐵} ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})) → ((𝑓‘∅) = (𝑓‘1o) → ∅ = 1o))
27	21, 23, 25, 26	syl12anc 1271	. . . . . . 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 6599	. . . . . . . 8 ⊢ 1o ≠ ∅
30	29	nesymi 2448	. . . . . . 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 625	. . . . 5 ⊢ (((𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} ∧ (𝑓‘∅) = 𝐵) ∧ (𝑓‘1o) = 𝐵) → 𝐴 ∈ V)
33		f1of 5583	. . . . . . . 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 5783	. . . . . . 7 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘1o) ∈ {𝐴, 𝐵})
36		elpri 3692	. . . . . . 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 805	. . . 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 5783	. . . . 5 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → (𝑓‘∅) ∈ {𝐴, 𝐵})
42		elpri 3692	. . . . 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 805	. . 3 ⊢ (𝑓:{∅, 1o}–1-1-onto→{𝐴, 𝐵} → 𝐴 ∈ V)
45	44	exlimiv 1646	. 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 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  ∅c0 3494  {cpr 3670   class class class wbr 4088  –1-1→wf1 5323  –1-1-onto→wf1o 5325  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575   ≈ cen 6906

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909

This theorem is referenced by:  pr2cv2  7400  pr2cv  7401