Theorem regsfromunir1 36670

Description: Derivation of ax-regs 35282 from unir1 9725. (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 9706	. . . . 5 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		fimass 6682	. . . . 5 ⊢ (rank:∪ (𝑅1 “ On)⟶On → (rank “ {𝑥 ∣ 𝜑}) ⊆ On)
3	1, 2	ax-mp 5	. . . 4 ⊢ (rank “ {𝑥 ∣ 𝜑}) ⊆ On
4		ffn 6662	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → rank Fn ∪ (𝑅1 “ On))
5	1, 4	ax-mp 5	. . . . . . 7 ⊢ rank Fn ∪ (𝑅1 “ On)
6		ssv 3958	. . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} ⊆ V
7		regsfromunir1.1	. . . . . . . 8 ⊢ ∪ (𝑅1 “ On) = V
8	6, 7	sseqtrri 3983	. . . . . . 7 ⊢ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)
9		fnimaeq0 6625	. . . . . . 7 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On)) → ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅))
10	5, 8, 9	mp2an 692	. . . . . 6 ⊢ ((rank “ {𝑥 ∣ 𝜑}) = ∅ ↔ {𝑥 ∣ 𝜑} = ∅)
11	10	necon3bii 2984	. . . . 5 ⊢ ((rank “ {𝑥 ∣ 𝜑}) ≠ ∅ ↔ {𝑥 ∣ 𝜑} ≠ ∅)
12	11	biimpri 228	. . . 4 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → (rank “ {𝑥 ∣ 𝜑}) ≠ ∅)
13		onint 7735	. . . 4 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank “ {𝑥 ∣ 𝜑}) ≠ ∅) → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
14	3, 12, 13	sylancr 587	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ → ∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}))
15		abn0 4337	. . 3 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
16		fvelimab 6906	. . . 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 692	. . 3 ⊢ (∩ (rank “ {𝑥 ∣ 𝜑}) ∈ (rank “ {𝑥 ∣ 𝜑}) ↔ ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
18	14, 15, 17	3imtr3i 291	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}))
19		vex 3444	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19, 7	eleqtrri 2835	. . . . . 6 ⊢ 𝑦 ∈ ∪ (𝑅1 “ On)
21		rankelb 9736	. . . . . 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 2825	. . . . . 6 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) ↔ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
24	23	biimpd 229	. . . . 5 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ((rank‘𝑧) ∈ (rank‘𝑦) → (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑})))
25		fnfvima 7179	. . . . . . . 8 ⊢ ((rank Fn ∪ (𝑅1 “ On) ∧ {𝑥 ∣ 𝜑} ⊆ ∪ (𝑅1 “ On) ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
26	5, 8, 25	mp3an12 1453	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑}))
27		onnmin 7743	. . . . . . 7 ⊢ (((rank “ {𝑥 ∣ 𝜑}) ⊆ On ∧ (rank‘𝑧) ∈ (rank “ {𝑥 ∣ 𝜑})) → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
28	3, 26, 27	sylancr 587	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} → ¬ (rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}))
29	28	con2i 139	. . . . 5 ⊢ ((rank‘𝑧) ∈ ∩ (rank “ {𝑥 ∣ 𝜑}) → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})
30	22, 24, 29	syl56 36	. . . 4 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
31	30	alrimiv 1928	. . 3 ⊢ ((rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
32	31	reximi 3074	. 2 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑦) = ∩ (rank “ {𝑥 ∣ 𝜑}) → ∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}))
33		df-rex 3061	. . 3 ⊢ (∃𝑦 ∈ {𝑥 ∣ 𝜑}∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})))
34		df-clab 2715	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
35		sb6 2090	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
36	34, 35	bitri 275	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
37		df-clab 2715	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
38		sb6 2090	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
39	37, 38	bitri 275	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
40	39	notbii 320	. . . . . . 7 ⊢ (¬ 𝑧 ∈ {𝑥 ∣ 𝜑} ↔ ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
41	40	imbi2i 336	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
42	41	albii 1820	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
43	36, 42	anbi12i 628	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑥 ∣ 𝜑})) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
44	43	exbii 1849	. . 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 219	. 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 18	1 ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780  [wsb 2067   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  ∪ cuni 4863  ∩ cint 4902   “ cima 5627  Oncon0 6317   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  𝑅₁cr1 9674  rankcrnk 9675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9676  df-rank 9677

This theorem is referenced by: (None)