Theorem onvf1odlem1 35285

Description: Lemma for onvf1od 35289. (Contributed by BTernaryTau, 2-Dec-2025.)

Assertion

Ref	Expression
onvf1odlem1	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem onvf1odlem1

Step	Hyp	Ref	Expression
1		nvel 5254	. . . . . 6 ⊢ ¬ V ∈ 𝑉
2		eleq1 2824	. . . . . . 7 ⊢ (V = 𝐴 → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	eqcoms 2744	. . . . . 6 ⊢ (𝐴 = V → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4	1, 3	mtbii 326	. . . . 5 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝑉)
5	4	con2i 139	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 = V)
6		eqv 3439	. . . . . 6 ⊢ (𝐴 = V ↔ ∀𝑦 𝑦 ∈ 𝐴)
7		alex 1828	. . . . . 6 ⊢ (∀𝑦 𝑦 ∈ 𝐴 ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
8	6, 7	bitri 275	. . . . 5 ⊢ (𝐴 = V ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
9	8	con2bii 357	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝐴 = V)
10	5, 9	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ¬ 𝑦 ∈ 𝐴)
11		ax-1 6	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝐴 → (𝑥 ∈ On → ¬ 𝑦 ∈ 𝐴))
12	11	ralrimiv 3128	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝐴 → ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
13	12	eximi 1837	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
14		rexv 3457	. . . 4 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
15	13, 14	sylibr 234	. . 3 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
16		tz9.13g 9716	. . . . 5 ⊢ (𝑦 ∈ V → ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
17	16	rgen 3053	. . . 4 ⊢ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)
18		r19.29r 3101	. . . . 5 ⊢ ((∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V (∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)))
19		r19.29 3100	. . . . . 6 ⊢ ((∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
20	19	reximi 3075	. . . . 5 ⊢ (∃𝑦 ∈ V (∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
21	18, 20	syl 17	. . . 4 ⊢ ((∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
22	17, 21	mpan2 692	. . 3 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
23	10, 15, 22	3syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
24		rexcom 3266	. . 3 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
25		exancom 1863	. . . . 5 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
26		rexv 3457	. . . . 5 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
27		df-rex 3062	. . . . 5 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
28	25, 26, 27	3bitr4i 303	. . . 4 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
29	28	rexbii 3084	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
30	24, 29	bitri 275	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
31	23, 30	sylib 218	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  Oncon0 6323  ‘cfv 6498  𝑅₁cr1 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688

This theorem is referenced by:  onvf1odlem2  35286  onvf1odlem4  35288