Theorem unxpdomlem3 8990

Description: Lemma for unxpdom 8991. (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 8987	. . 3 ⊢ (𝑎 ∈ V → (1o ≺ 𝑎 ↔ ∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛))
2	1	elv 3436	. 2 ⊢ (1o ≺ 𝑎 ↔ ∃𝑚 ∈ 𝑎 ∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛)
3		1sdom 8987	. . 3 ⊢ (𝑏 ∈ V → (1o ≺ 𝑏 ↔ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡))
4	3	elv 3436	. 2 ⊢ (1o ≺ 𝑏 ↔ ∃𝑠 ∈ 𝑏 ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡)
5		reeanv 3294	. . 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 3294	. . . . 5 ⊢ (∃𝑛 ∈ 𝑎 ∃𝑡 ∈ 𝑏 (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡))
7		vex 3434	. . . . . . . . 9 ⊢ 𝑎 ∈ V
8		vex 3434	. . . . . . . . 9 ⊢ 𝑏 ∈ V
9	7, 8	unex 7587	. . . . . . . 8 ⊢ (𝑎 ∪ 𝑏) ∈ V
10	7, 8	xpex 7594	. . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
13		simp2r 1198	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡 ∈ 𝑏)
14		simp1r 1196	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠 ∈ 𝑏)
15	13, 14	ifcld 4510	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
16	15	ad2antrr 722	. . . . . . . . . . . . 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 5626	. . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛 ∈ 𝑎)
19		simp1l 1195	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚 ∈ 𝑎)
20	18, 19	ifcld 4510	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
21	20	ad2antrr 722	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) → 𝑥 ∈ (𝑎 ∪ 𝑏))
23		elun 4087	. . . . . . . . . . . . . . 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 999	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎 ∪ 𝑏)) ∧ ¬ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑏)
26	21, 25	opelxpd 5626	. . . . . . . . . . . 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 4499	. . . . . . . . . . 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 2844	. . . . . . . . . 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 6982	. . . . . . . . 9 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎 ∪ 𝑏)⟶(𝑎 × 𝑏))
31	29, 11	unxpdomlem1 8988	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
32	31	ad2antrl 724	. . . . . . . . . . . . . . 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 4470	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 8988	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑤) = if(𝑤 ∈ 𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
37	36	ad2antll 725	. . . . . . . . . . . . . . 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 4470	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 2755	. . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
43		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
44		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑠 ∈ V
45	43, 44	ifex 4514	. . . . . . . . . . . . . 14 ⊢ if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
46	42, 45	opth1 5392	. . . . . . . . . . . . 13 ⊢ (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
47	41, 46	syl6bi 252	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
48		simprr 769	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → 𝑤 ∈ (𝑎 ∪ 𝑏))
49		simpll 763	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ¬ 𝑚 = 𝑛)
50		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ¬ 𝑠 = 𝑡)
51	29, 11, 48, 49, 50	unxpdomlem2 8989	. . . . . . . . . . . . 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 2746	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑤) = (𝐹‘𝑧))
54		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → 𝑧 ∈ (𝑎 ∪ 𝑏))
55	29, 11, 54, 49, 50	unxpdomlem2 8989	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (𝑤 ∈ 𝑎 ∧ ¬ 𝑧 ∈ 𝑎)) → ¬ (𝐹‘𝑤) = (𝐹‘𝑧))
56	55	ancom2s 646	. . . . . . . . . . . . . 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	syl5bi 241	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) ∧ (¬ 𝑧 ∈ 𝑎 ∧ 𝑤 ∈ 𝑎)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
59		iffalse 4473	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 4473	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 2755	. . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ V
67		vex 3434	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
68	66, 67	ifex 4514	. . . . . . . . . . . . . . 15 ⊢ if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
69	68, 42	opth 5393	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . 13 ⊢ (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
71	65, 70	syl6bi 252	. . . . . . . . . . . 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 1038	. . . . . . . . . . 11 ⊢ (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎 ∪ 𝑏) ∧ 𝑤 ∈ (𝑎 ∪ 𝑏))) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
73	72	ralrimivva 3116	. . . . . . . . . 10 ⊢ ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎 ∪ 𝑏)∀𝑤 ∈ (𝑎 ∪ 𝑏)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
74	73	3ad2ant3 1133	. . . . . . . . 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 7122	. . . . . . . . 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 582	. . . . . . . 8 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎 ∪ 𝑏)–1-1→(𝑎 × 𝑏))
77		f1dom2g 8728	. . . . . . . 8 ⊢ (((𝑎 ∪ 𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎 ∪ 𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
78	9, 10, 76, 77	mp3an12i 1463	. . . . . . 7 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
79	78	3expia 1119	. . . . . 6 ⊢ (((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) ∧ (𝑛 ∈ 𝑎 ∧ 𝑡 ∈ 𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
80	79	rexlimdvva 3224	. . . . 5 ⊢ ((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) → (∃𝑛 ∈ 𝑎 ∃𝑡 ∈ 𝑏 (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
81	6, 80	syl5bir 242	. . . 4 ⊢ ((𝑚 ∈ 𝑎 ∧ 𝑠 ∈ 𝑏) → ((∃𝑛 ∈ 𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡 ∈ 𝑏 ¬ 𝑠 = 𝑡) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)))
82	81	rexlimivv 3222	. . 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 597	1 ⊢ ((1o ≺ 𝑎 ∧ 1o ≺ 𝑏) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066  Vcvv 3430   ∪ cun 3889  ifcif 4464  ⟨cop 4572   class class class wbr 5078   ↦ cmpt 5161   × cxp 5586  ⟶wf 6426  –1-1→wf1 6427  ‘cfv 6430  1_oc1o 8274   ≼ cdom 8705   ≺ csdm 8706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1o 8281  df-2o 8282  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711

This theorem is referenced by:  unxpdom  8991