Theorem unxpdomlem3 9073

Description: Lemma for unxpdom 9074. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Hypotheses

Ref	Expression
unxpdomlem1.1	⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
unxpdomlem1.2	⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)

Assertion

Ref	Expression
unxpdomlem3	⊢ ((1o ≺ 𝑎 ∧ 1o ≺ 𝑏) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))

Distinct variable group:   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem3

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1sdom 9069	. . 3 ⊢ (𝑎 ∈ V → (1o ≺ 𝑎 ↔ ∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛))
2	1	elv 3443	. 2 ⊢ (1o ≺ 𝑎 ↔ ∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛)
3		1sdom 9069	. . 3 ⊢ (𝑏 ∈ V → (1o ≺ 𝑏 ↔ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡))
4	3	elv 3443	. 2 ⊢ (1o ≺ 𝑏 ↔ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡)
5		reeanv 3214	. . 3 ⊢ (∃𝑚 ∈ 𝑎 ∃𝑠 ∈ 𝑏 (∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡) ↔ (∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡))
6		reeanv 3214	. . . . 5 ⊢ (∃𝑛 ∈ 𝑎 ∃𝑡 ∈ 𝑏 (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡))
7		vex 3441	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		vex 3441	. . . . . . . . 9 ⊢ 𝑏 ∈ V
9	7, 8	unex 7628	. . . . . . . 8 ⊢ (𝑎 ∪ 𝑏) ∈ V
10	7, 8	xpex 7635	. . . . . . . 8 ⊢ (𝑎 × 𝑏) ∈ V
11		unxpdomlem1.2	. . . . . . . . . . 11 ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
12		simpr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
13		simp2r 1200	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡 ∈ 𝑏)
14		simp1r 1198	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠 ∈ 𝑏)
15	13, 14	ifcld 4511	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
16	15	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ 𝑥 ∈ 𝑎) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
17	12, 16	opelxpd 5638	. . . . . . . . . . . 12 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ 𝑥 ∈ 𝑎) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
18		simp2l 1199	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛 ∈ 𝑎)
19		simp1l 1197	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚 ∈ 𝑎)
20	18, 19	ifcld 4511	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
21	20	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ ¬ 𝑥 ∈ 𝑎) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
22		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) → 𝑥 ∈ (𝑎 ∪ 𝑏))
23		elun 4089	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑎 ∪ 𝑏) ↔ (𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏))
24	22, 23	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) → (𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏))
25	24	orcanai 1001	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ ¬ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑏)
26	21, 25	opelxpd 5638	. . . . . . . . . . . 12 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ ¬ 𝑥 ∈ 𝑎) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
27	17, 26	ifclda 4500	. . . . . . . . . . 11 ⊢ ((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) → if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩) ∈ (𝑎 × 𝑏))
28	11, 27	eqeltrid 2841	. . . . . . . . . 10 ⊢ ((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) → 𝐺 ∈ (𝑎 × 𝑏))
29		unxpdomlem1.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
30	28, 29	fmptd 7020	. . . . . . . . 9 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎 ∪ 𝑏)⟶(𝑎 × 𝑏))
31	29, 11	unxpdomlem1 9071	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
32	31	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
33		iftrue 4471	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑎 → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
34	33	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎) → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
35	32, 34	sylan9eq 2796	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → (𝐹‘𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
36	29, 11	unxpdomlem1 9071	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑤) = if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
37	36	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → (𝐹‘𝑤) = if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
38		iftrue 4471	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝑎 → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
39	38	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎) → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
40	37, 39	sylan9eq 2796	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → (𝐹‘𝑤) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
41	35, 40	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩))
42		vex 3441	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
43		vex 3441	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
44		vex 3441	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
45	43, 44	ifex 4515	. . . . . . . . . . . . . 14 ⊢ if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
46	42, 45	opth1 5403	. . . . . . . . . . . . 13 ⊢ (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
47	41, 46	syl6bi 253	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
48		simprr 771	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → 𝑤 ∈ (𝑎 ∪ 𝑏))
49		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ¬ 𝑚 = 𝑛)
50		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ¬ 𝑠 = 𝑡)
51	29, 11, 48, 49, 50	unxpdomlem2 9072	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))
52	51	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
53		eqcom 2743	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑤) = (𝐹‘𝑧))
54		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → 𝑧 ∈ (𝑎 ∪ 𝑏))
55	29, 11, 54, 49, 50	unxpdomlem2 9072	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑤 ∈ 𝑎 ∧ ¬ 𝑧 ∈ 𝑎)) → ¬ (𝐹‘𝑤) = (𝐹‘𝑧))
56	55	ancom2s 648	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑤) = (𝐹‘𝑧))
57	56	pm2.21d 121	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑤) = (𝐹‘𝑧) → 𝑧 = 𝑤))
58	53, 57	biimtrid 241	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
59		iffalse 4474	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧 ∈ 𝑎 → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
60	59	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎) → if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
61	32, 60	sylan9eq 2796	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑧) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
62		iffalse 4474	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑤 ∈ 𝑎 → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
63	62	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎) → if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
64	37, 63	sylan9eq 2796	. . . . . . . . . . . . . 14 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → (𝐹‘𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
65	61, 64	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
66		vex 3441	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ V
67		vex 3441	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
68	66, 67	ifex 4515	. . . . . . . . . . . . . . 15 ⊢ if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
69	68, 42	opth 5404	. . . . . . . . . . . . . 14 ⊢ (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ ↔ (if(𝑧 = 𝑡, 𝑛, 𝑚) = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ 𝑧 = 𝑤))
70	69	simprbi 498	. . . . . . . . . . . . 13 ⊢ (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
71	65, 70	syl6bi 253	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
72	47, 52, 58, 71	4casesdan 1040	. . . . . . . . . . 11 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
73	72	ralrimivva 3194	. . . . . . . . . 10 ⊢ ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎 ∪ 𝑏)∀𝑤 ∈ (𝑎 ∪ 𝑏)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
74	73	3ad2ant3 1135	. . . . . . . . 9 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → ∀𝑧 ∈ (𝑎 ∪ 𝑏)∀𝑤 ∈ (𝑎 ∪ 𝑏)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
75		dff13 7160	. . . . . . . . 9 ⊢ (𝐹:(𝑎 ∪ 𝑏)–1-1→(𝑎 × 𝑏) ↔ (𝐹:(𝑎 ∪ 𝑏)⟶(𝑎 × 𝑏) ∧ ∀𝑧 ∈ (𝑎 ∪ 𝑏)∀𝑤 ∈ (𝑎 ∪ 𝑏)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
76	30, 74, 75	sylanbrc 584	. . . . . . . 8 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎 ∪ 𝑏)–1-1→(𝑎 × 𝑏))
77		f1dom2g 8790	. . . . . . . 8 ⊢ (((𝑎 ∪ 𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎 ∪ 𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
78	9, 10, 76, 77	mp3an12i 1465	. . . . . . 7 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
79	78	3expia 1121	. . . . . 6 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
80	79	rexlimdvva 3202	. . . . 5 ⊢ ((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) → (∃𝑛 ∈ 𝑎 ∃𝑡 ∈ 𝑏 (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
81	6, 80	syl5bir 243	. . . 4 ⊢ ((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) → ((∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
82	81	rexlimivv 3193	. . 3 ⊢ (∃𝑚 ∈ 𝑎 ∃𝑠 ∈ 𝑏 (∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
83	5, 82	sylbir 234	. 2 ⊢ ((∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
84	2, 4, 83	syl2anb 599	1 ⊢ ((1o ≺ 𝑎 ∧ 1o ≺ 𝑏) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071  Vcvv 3437   ∪ cun 3890  ifcif 4465  ⟨cop 4571   class class class wbr 5081   ↦ cmpt 5164   × cxp 5598  ⟶wf 6454  –1-1→wf1 6455  ‘cfv 6458  1_oc1o 8321   ≼ cdom 8762   ≺ csdm 8763

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-1o 8328  df-2o 8329  df-en 8765  df-dom 8766  df-sdom 8767

This theorem is referenced by:  unxpdom  9074