Theorem regsfromunir1 37084

Description: Derivation of ax-regs 35555 from unir1 9788. (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 9769	. . . . 5 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		fimass 6730	. . . . 5 ⊢ (rank:∪ (𝑅1 “ On)⟶On → (rank “ {𝑥 ∣ 𝜑}) ⊆ On)
3	1, 2	ax-mp 5	. . . 4 ⊢ (rank “ {𝑥 ∣ 𝜑}) ⊆ On
4		ffn 6709	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5	1, 4	ax-mp 5	. . . . . . 7 ⊢ rank Fn ∪ (𝑅1 “ On)
6		ssv 3962	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} ⊆ V
7		regsfromunir1.1	. . . . . . . 8 ⊢ ∪ (𝑅1 “ On) = V
8	6, 7	sseqtrri 3987	. . . . . . 7 ⊢ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)
9		fnimaeq0 6672	. . . . . . 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 3012	. . . . 5 ⊢ ((rank “ {𝑥 ∣ 𝜑}) ≠ ∅ ↔ {𝑥 ∣ 𝜑} ≠ ∅)
12	11	biimpri 231	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → (rank “ {𝑥 ∣ 𝜑}) ≠ ∅)
13		onint 7791	. . . 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 4341	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
16		fvelimab 6957	. . . 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 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19, 7	eleqtrri 2864	. . . . . 6 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On)
21		rankelb 9799	. . . . . 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 2854	. . . . . 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 7235	. . . . . . . 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 7799	. . . . . . 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 1960	. . 3 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
32	31	reximi 3105	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
33		df-rex 3092	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})))
34		df-clab 2744	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
35		sb6 2122	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
36	34, 35	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
37		df-clab 2744	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
38		sb6 2122	. . . . . . . . 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 1852	. . . . 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 1881	. . 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 1568   = wceq 1570  ∃wex 1812  [wsb 2099   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ∪ cuni 4874  ∩ cint 4914   “ cima 5666  Oncon0 6364   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  𝑅₁cr1 9737  rankcrnk 9738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9739  df-rank 9740

This theorem is used by: (None)