Theorem onfrALT 44574

Description: The membership relation is foundational on the class of ordinal numbers. onfrALT 44574 is an alternate proof of onfr 6391. onfrALTVD 44915 is the Virtual Deduction proof from which onfrALT 44574 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6391 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 44915. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALT	⊢ E Fr On

Proof of Theorem onfrALT

Dummy variables 𝑥 𝑎 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfepfr 5638	. 2 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
2		simpr 484	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3		n0 4328	. . . 4 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎)
4		onfrALTlem1 44573	. . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
5	4	expd 415	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
6		onfrALTlem2 44571	. . . . . . 7 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
7	6	expd 415	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
8		pm2.61 192	. . . . . 6 ⊢ (((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
9	5, 7, 8	syl6c 70	. . . . 5 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
10	9	exlimdv 1933	. . . 4 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
11	3, 10	biimtrid 242	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
12	2, 11	mpd 15	. 2 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
13	1, 12	mpgbir 1799	1 ⊢ E Fr On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ≠ wne 2932  ∃wrex 3060   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308   E cep 5552   Fr wfr 5603  Oncon0 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2376  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355  df-on 6356

This theorem is referenced by: (None)