Theorem regsfromunir1 37421

Description: Derivation of ax-regs 35851 from unir1 9825. (Contributed by Matthew House, 4-Mar-2026.)

Hypothesis

Ref	Expression
regsfromunir1.1	⊢ ∪ (𝑅1 “ On) = V

Assertion

Ref	Expression
regsfromunir1	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem regsfromunir1

Step	Hyp	Ref	Expression
1		rankf 9806	. . . . 5 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		fimass 6731	. . . . 5 ⊢ (rank:∪ (𝑅1 “ On)⟶On → (rank “ {𝑥 ∣ 𝜑}) ⊆ On)
3	1, 2	ax-mp 5	. . . 4 ⊢ (rank “ {𝑥 ∣ 𝜑}) ⊆ On
4		ffn 6710	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5	1, 4	ax-mp 5	. . . . . . 7 ⊢ rank Fn ∪ (𝑅1 “ On)
6		ssv 3932	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} ⊆ V
7		regsfromunir1.1	. . . . . . . 8 ⊢ ∪ (𝑅1 “ On) = V
8	6, 7	sseqtrri 3957	. . . . . . 7 ⊢ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)
9		fnimaeq0 6673	. . . . . . 7 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅))
10	5, 8, 9	mp2an 705	. . . . . 6 ⊢ ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅)
11	10	necon3bii 3007	. . . . 5 ⊢ ((rank “ {𝑥 ∣ 𝜑}) ≠ ∅ ↔ {𝑥 ∣ 𝜑} ≠ ∅)
12	11	biimpri 231	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → (rank “ {𝑥 ∣ 𝜑}) ≠ ∅)
13		onint 7805	. . . 4 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank “ {𝑥 ∣ 𝜑}) ≠ ∅) → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
14	3, 12, 13	sylancr 599	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
15		abn0 4333	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
16		fvelimab 6958	. . . 4 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → (∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}) ↔ ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑})))
17	5, 8, 16	mp2an 705	. . 3 ⊢ (∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}) ↔ ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
18	14, 15, 17	3imtr3i 294	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
19		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19, 7	eleqtrri 2859	. . . . . 6 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On)
21		rankelb 9837	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑧 ∈ 𝑦 → (rank‘𝑧) ∈ (rank‘𝑦)))
22	20, 21	ax-mp 5	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → (rank‘𝑧) ∈ (rank‘𝑦))
23		eleq2 2849	. . . . . 6 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) ↔ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
24	23	biimpd 232	. . . . 5 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) → (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
25		fnfvima 7240	. . . . . . . 8 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On) ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
26	5, 8, 25	mp3an12 1480	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
27		onnmin 7813	. . . . . . 7 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑})) → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
28	3, 26, 27	sylancr 599	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
29	28	con2i 140	. . . . 5 ⊢ ((rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}) → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})
30	22, 24, 29	syl56 37	. . . 4 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
31	30	alrimiv 1959	. . 3 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
32	31	reximi 3100	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
33		df-rex 3087	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})))
34		df-clab 2739	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
35		sb6 2121	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
36	34, 35	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
37		df-clab 2739	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
38		sb6 2121	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
39	37, 38	bitri 278	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
40	39	notbii 323	. . . . . . 7 ⊢ (¬ 𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
41	40	imbi2i 339	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
42	41	albii 1851	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
43	36, 42	anbi12i 640	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
44	43	exbii 1880	. . 3 ⊢ (∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
45	33, 44	sylbb 222	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
46	18, 32, 45	3syl 19	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811  [wsb 2098   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ∪ cuni 4866  ∩ cint 4906   “ cima 5653  Oncon0 6362   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  𝑅₁cr1 9770  rankcrnk 9771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-r1 9772  df-rank 9773

This theorem is used by: (None)