Theorem ordsucunielexmid 4532

Description: The converse of sucunielr 4511 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)

Hypothesis

Ref	Expression
ordsucunielexmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)

Assertion

Ref	Expression
ordsucunielexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ordsucunielexmid

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eloni 4377	. . . . . . . 8 ⊢ (𝑏 ∈ On → Ord 𝑏)
2		ordtr 4380	. . . . . . . 8 ⊢ (Ord 𝑏 → Tr 𝑏)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝑏 ∈ On → Tr 𝑏)
4		vex 2742	. . . . . . . 8 ⊢ 𝑏 ∈ V
5	4	unisuc 4415	. . . . . . 7 ⊢ (Tr 𝑏 ↔ ∪ suc 𝑏 = 𝑏)
6	3, 5	sylib 122	. . . . . 6 ⊢ (𝑏 ∈ On → ∪ suc 𝑏 = 𝑏)
7	6	eleq2d 2247	. . . . 5 ⊢ (𝑏 ∈ On → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
8	7	adantl 277	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 ↔ 𝑎 ∈ 𝑏))
9		onsuc 4502	. . . . 5 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
10		ordsucunielexmid.1	. . . . . 6 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)
11		eleq1 2240	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ 𝑦))
12		suceq 4404	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
13	12	eleq1d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑎 ∈ 𝑦))
14	11, 13	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦) ↔ (𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦)))
15		unieq 3820	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑏 → ∪ 𝑦 = ∪ suc 𝑏)
16	15	eleq2d 2247	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (𝑎 ∈ ∪ 𝑦 ↔ 𝑎 ∈ ∪ suc 𝑏))
17		eleq2 2241	. . . . . . . 8 ⊢ (𝑦 = suc 𝑏 → (suc 𝑎 ∈ 𝑦 ↔ suc 𝑎 ∈ suc 𝑏))
18	16, 17	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = suc 𝑏 → ((𝑎 ∈ ∪ 𝑦 → suc 𝑎 ∈ 𝑦) ↔ (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏)))
19	14, 18	rspc2va 2857	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ suc 𝑏 ∈ On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
20	10, 19	mpan2 425	. . . . 5 ⊢ ((𝑎 ∈ On ∧ suc 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
21	9, 20	sylan2 286	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ ∪ suc 𝑏 → suc 𝑎 ∈ suc 𝑏))
22	8, 21	sylbird 170	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏))
23	22	rgen2a 2531	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On (𝑎 ∈ 𝑏 → suc 𝑎 ∈ suc 𝑏)
24	23	onsucelsucexmid 4531	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∪ cuni 3811  Tr wtr 4103  Ord word 4364  Oncon0 4365  suc csuc 4367

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by: (None)